Gerwin Griffioen's Technical Analysis Pages

Griffioen, G.A.W. (2003), "Technical Analysis in Financial Markets", PhD thesis, University of Amsterdam.

Chapter 3
Technical Trading Rule Performance in Dow-Jones Industrial Average Listed Stocks

3.1 Introduction

In 1882 Charles H. Dow, Edward D. Jones and Charles M. Bergstresser started Dow, Jones & Co., publisher of the ``Customer's Afternoon Letter''. This was the precursor of ``The Wall Street Journal'', which was founded in 1889. In those early days trading was dominated by pools and prices were subject to spectacular rises and declines. Trading was mainly done on inside information. Stocks were considered to be for gamblers, raiders and speculators. Charles Dow discerned three types of market movements. First there are the daily actions, which reflect speculators' activities, called tertiary or minor trends. Second there are the secondary or intermediate trends, that is short swings of two weeks to a month or more, which reflect the strategies of large investment pools. Charles Dow considered the first two movements to be the result of market manipulations and he advised not to become involved with any kind of speculation, because he believed this was a sure way to lose money. Third, he discerned four-year movements, the primary or major trend, derived from economic forces beyond the control of individuals. Charles Dow thought that expectations for the national economy were translated into market orders that caused stock prices to rise or fall over the long term - usually in advance of actual economic developments. He believed that fundamental economic variables determine prices in the long run. To quantify his theory Charles Dow began to compute averages to measure market movements.

In 1884 Charles Dow started to construct an average of eleven stocks, composed of nine railroad companies and only two non-railroad companies, because in those days railroad companies were the first large national corporations. He recognized that railroad companies presented only a partial picture of the economy and that industrial companies were crucial contributors to America's growth. ``What the industrials make the railroads take'' was his slogan and from this he concluded that two separate measures could act as coconfirmers to detect any broad market trend. This idea led to the birth of the Dow-Jones Railroad Average (DJRA), renamed in 1970 to Dow-Jones Transportation Average, and to the birth of the Dow-Jones Industrial Average (DJIA). The DJIA started on May 26, 1896 at 40.94 points and the DJRA started on September 8, 1896 at 48.55 points.

Initially the DJIA contained 12 stocks. This number was increased to 20 in 1916 and on October 1, 1928 the index was expanded to a 30-stock average, which it still is. The only company permanently present in the index, except for a break between 1898-1907, is General Electric. The first 25 years of its existence the DJIA was not yet known among a wide class of people. In the roaring twenties the DJIA got its popularity, when masses of average citizens began buying stocks. It became a tool by which the general public could measure the overall performance of the US stock market and it gave investors a sense of what was happening in this market. After the crash of 1929 the DJIA made front-page headlines to measure the overall damage in personal investments. The DJIA has been published continuously for more than one hundred years, except for four and a half months at the beginning of World War I when the New York Stock Exchange (NYSE) closed temporarily. Nowadays the DJIA is the oldest and most famous measure of the US stock market.

The DJIA is price weighted rather than market weighted, because of the technology in Charles Dow's days. It is an equally-weighted price average of 30 blue-chip US stocks, each of them representing a particular industry. When stocks split or when the DJIA is revised by excluding and including certain stocks, the divisor is updated to preserve historical continuity. Because the composition of the DJIA is dependent on the decision which stocks to exclude and to include, the index would have a completely different value today, if the DJIA constructors had made different decisions in the past. People criticize the Dow because it is too narrow. It only contains 30 stocks out of thousands of public companies and the calculation is simplistic. However it has been shown that the DJIA tracks other major market indices fairly closely. It follows closely the movement of market-weighted indices such as the NASDAQ composite, NYSE composite, Russell 2000, Standard & Poor's 500 and the Wilshire 5000 (Prestbo, 1999, p.47).

It was William Peter Hamilton in his book ``The Stock Market Barometer'' (1922) who laid the foundation of ``the Dow Theory'', the first theory of chart readers. The theory is based on editorials of Charles H. Dow when he was editor of the Wall Street Journal in the period 1889-1902. Robert Rhea popularized the idea in his 1930s market letters and his book ``The Dow Theory'' (1932). Although the theory bears Charles Dow's name, it is likely that he would deny any allegiance to it. Instead of being a chartist, Charles Dow as a financial reporter advocated to invest on sound fundamental economic variables, that is buying stocks when their prices are well below their fundamental values. His main purpose in developing the averages was to measure market cycles, rather than to use them to generate trading signals.

After the work of Hamilton and Rhea the technical analysis literature was expanded and refined by Richard Schabacker, Robert Edwards and John Magee, and later by Welles Wilder and John Murphy. Technical analysis developed itself into a standard tool used by many to forecast the future price path of all kinds of financial assets such as stocks, bonds, futures and options. Nowadays a lot of technical analysis software packages are sold on the market. Technical analysis newsletters and journals flourish. Every bank employs several chartists who write technical reports spreading around forecasts with all kinds of fancy techniques. Classes (also through the internet) are organized to introduce the home investor in the topic. Technical analysis has become an industry on its own. For example, the questionnaire surveys of Taylor and Allen (1992), Menkhoff (1998) and Cheung and Chinn (1999) show that technical analysis is broadly used in practice. However, despite the fact that chartists have a strong belief in their forecasting ability, for academics it remains the question whether it has any statistically significant forecasting power and whether it can be profitably exploited also after accounting for transaction costs and risk.

Cowles (1933) considered the 26-year forecasting record of Hamilton in the period 1903-1929. He found that Hamilton could not beat a continuous investment in the DJIA or the DJRA after correcting for the effect of brokerage charges, cash dividends and interest earned when not in the market. On 90 occasions Hamilton announced changes in the outlook for the market. It was found that 45 of his changes of position were unsuccessful and that 45 were successful. In a later period, Alexander (1964), and Fama and Blume (1966) found that filter strategies, intended to reveal possible trends in the data, did not yield profits after correcting for transaction costs, when applied to the DJIA and to individual stocks that composed the DJIA. The influential paper of Fama (1970) reviews the theoretical and empirical literature on the efficient markets model until that date and concludes that the evidence in support of the efficient markets model is very extensive, and that contradictory evidence is sparse. From that moment on the efficient markets hypothesis (EMH), which states that it is not possible to forecast the future price movements of a financial asset given any information set, is the central paradigm in financial economics. The impact Fama's (1970) paper was so large, that it took a while before new academic literature on technical trading was published.

The extensive study of Brock, Lakonishok and LeBaron (1992) on technical analysis led to a renewed interest in the topic. They applied 26 simple technical trading strategies, such as moving averages, and support-and-resistance strategies, to the daily closing prices of the DJIA in the period 1897-1986, nearly 90 years of data. They were the first who extended simple standard statistical analysis with parametric bootstrap techniques, inspired by Efron (1979), Freedman and Peters (1984a, 1984b), and Efron and Tibshirani (1986). It was found that the predictive ability of the technical trading rules found was not consistent with a random walk, an AR(1), a GARCH-in-mean model, or an exponential GARCH. The strong results of Brock et al. (1992) were the impetus for many papers published on technical analysis in the 1990s.

Although numerous papers found evidence for economic profitability and statistically significant forecasting power of technical trading rules, they did acknowledge the problem of data snooping. This is the danger that the results of the best forecasting rule may just be generated by chance, instead of truly superior forecasting power over the buy-and-hold benchmark. It could be that the trading rules under consideration were the result of survivorship bias. That is, the best trading rules found by chartists in the past get most attention by academic researchers in the present. Finally White (2000), building on the work of Diebold and Mariano (1995) and West (1996), developed a simple and straightforward procedure, called the Reality Check (RC), for testing the null hypothesis that the best model encountered in a specification search has no predictive superiority over a given benchmark model. Sullivan, Timmermann and White (1999) utilize the RC to evaluate a large set of approximately 7800 simple technical trading strategies on the data set of Brock et al. (1992). They confirm that the results found by Brock et al. (1992) still hold after correcting for data snooping. However in the out-of-sample period 1986-1996 they find no significant forecasting ability for the technical trading strategies anymore. Hansen (2001) shows that the RC is a biased test, which yields inconsistent p-values. Moreover, the test is sensitive to the inclusion of poor and irrelevant models. Further the test has poor power properties, which can be driven to zero. Therefore, within the framework of White (2000), Hansen (2001) derives a test for superior predictive ability (SPA).

In this chapter we test whether objective computerized trend-following technical trading techniques can profitably be exploited after correction for transaction costs when applied to the DJIA and to all stocks listed in the DJIA in the period 1973:1-2001:6. Furthermore, we test whether the best strategies can beat the buy-and-hold benchmark significantly after correction for data snooping. This chapter may be seen as an empirical application of White's RC and Hansen's SPA-test. In addition we test by recursively optimizing our trading rule set whether technical analysis shows true out-of-sample forecasting power.

In section 3.2 we list the stock price data examined in this chapter and we show the summary statistics. Section 3.3 presents an overview of the technical trading rules applied to the stock price data. Section 3.4 describes which performance measures are used and how they are calculated. In section 3.5 the problem of data snooping is addressed and a short summary of White's RC and Hansen's SPA-test is presented. Section 3.6 shows the empirical results. In section 3.7 we test whether recursively optimizing and updating our technical trading rule set shows genuine out-of-sample forecasting ability. Finally section 3.8 concludes.

3.2 Data and summary statistics

The data series examined in this chapter are the daily closing levels of the Dow-Jones Industrial Average (DJIA) and the daily closing stock prices of 34 companies listed in the DJIA in the period January 2, 1973 through June 29, 2001. Table 3.1 lists the data series. The companies in the DJIA are the largest and most important in their industries. Prices are corrected for dividends, capital changes and stock splits. As a proxy for the risk-free interest rate we use daily data on US 3-month certificates of deposits. Several studies found that technical trading rules show significant forecasting power in the era until 1987 and no forecasting power anymore from then onwards. Therefore we split our data sample in two subperiods. Table 3.2 shows the summary statistics for the period 1973-2001 and the tables 3.3 and 3.4 show the summary statistics for the two subperiods 1973-1986 and 1987-2001. Because the first 260 data points are used for initializing the technical trading strategies, the summary statistics are shown from January 1, 1974. In the tables the first and second column show the names of the data series examined and the number of available data points. The third column shows the mean yearly effective return in percentage/100 terms. The fourth through seventh column show the mean, standard deviation, skewness and kurtosis of the logarithmic daily return. The eight column shows the t-ratio to test whether the mean logarithmic return is significantly different from zero. The ninth column shows the Sharpe ratio, that is the extra return over the risk-free interest rate per extra point of risk, as measured by the standard deviation. The tenth column shows the largest cumulative loss, that is the largest decline from a peak to a through, of the data series in percentage/100 terms. The eleventh column shows the Ljung-Box (1978) Q-statistic testing whether the first 20 autocorrelations of the return series as a whole are significantly different from zero. The twelfth column shows the heteroskedasticity adjusted Box-Pierce (1970) Q-statistic, as derived by Diebold (1986). The final column shows the Ljung-Box (1978) Q-statistic testing for autocorrelations in the squared returns.

All data series, except Bethlehem Steel, show in the full sample period a positive mean yearly return which is on average 11.5%. The return distributions are strongly leptokurtic and show signs of negative skewness, especially for the DJIA, Eastman Kodak and Procter & Gamble. The 34 separate stocks are riskier than the index, which is shown by the standard deviation of the returns. On average it is 1.9% for the 34 stocks, while it is 1% for the DJIA. Thus it is clear that firm specific risks are reduced by a diversified index. The Sharpe ratio is negative for 12 stocks, which means that these stocks were not able to beat a continuous risk free investment. Table 3.1 shows that the largest decline of the DJIA is equal to 36% and took place in the period August 26, 1987 until October 19, 1987 that covers the crash of 1987. October 19, 1987 showed the biggest one-day percentage loss in history of the DJIA and brought the index down by 22.61%. October 21, 1987 on its turn showed the largest one-day gain and brought the index up by 9.67%. However the largest decline of each of the 34 separate stocks is larger, on average 61%. For only five stocks (GoodYear Tire, HP, Home Depot, IBM, Wal-Mart) the largest decline started around August 1987. As can be seen in the table, the increasing oil prices during the seventies, caused initially by the oil embargo of the Arab oil exporting countries against countries supporting Israel in ``The Yom Kippur War'' in 1973, had the largest impact on stock prices. The doubling of oil prices led to a widespread recession and a general crisis of confidence. Bethlehem Steel did not perform very well during the entire 1973-2001 period and declined 97% during the largest part of its sample. AT&T declined 73% within two years: February 4, 1999 until December 28, 2000 which covers the so-called burst of the internet and telecommunications bubble.

If the summary statistics of the two subperiods in tables 3.3 and 3.4 are compared, then some substantial differences can be noticed. The mean yearly return of the DJIA is in the first subperiod 1973-1986 equal to 6.1%, while in the second subperiod 1987-2001 it is equal to 12.1%, almost twice as large. For almost all data series the standard deviation of the returns is higher in the second subperiod than in the first subperiod. The Sharpe ratio is negative for only 5 stocks in the subperiod 1987-2001, while it is negative for 22 stocks and the DJIA in the period 1973-1986, clearly indicating that buy-and-hold stock investments had a hard time in beating a risk free investment particularly in the first subperiod. Also in the second subperiod the return distributions are strongly leptokurtic and negatively skewed, which stands in contrast with the first subperiod, where the kurtosis of the return distributions is much lower and where the skewness is slightly positive for most stocks. Thus, large one-day price changes, especially negative ones, occur more often in the second than in the first subperiod. Higher rewards of holding stocks in the second subperiod come together with higher risks.

We computed autocorrelation functions (ACFs) of the returns and significance is tested with Bartlett (1946) standard errors and Diebold's (1986) heteroskedasticity-consistent standard errors¹. Under the assumption that the data is white noise with constant variance the standard error for each sample autocorrelation is equal to 1/n. However Hsieh (1988) points out that sample autocorrelation may be spurious in the presence of heteroskedasticity, because the standard error of each sample autocorrelation may be underestimated by 1/n. Diebold's (1986) heteroskedasticity-consistent estimate of the standard error for the k-th sample autocorrelation, r_k, is calculated as follows:

s.e.(r_k)=1/n (1+g(r²,k)/s⁴) ,

where g(r²,k) is the k-th order sample autocorrelation function of the squared returns, and s is the sample standard deviation of the returns. Moreover Diebold (1986) showed that the adjusted Box-Pierce (1970) Q-statistic

�

k=1

�
�
�
�

r_k

s.e.(r_k)

�
�
�
�

to test that the first q autocorrelations as a whole are not significantly different from zero, is asymptotically c-squared distributed with q degrees of freedom. Typically autocorrelations of the returns are small with only few lags being significant. It is noteworthy that for most data series the second order autocorrelation is negative in all periods. The first order autocorrelation is negative for only 3 data series in the period 1973-1986, while it is negative for 18 data series in the period 1987-2001. The Ljung-Box (1978) Q-statistics in the second to last columns of tables 3.2, 3.3 and 3.4 reject for all periods for almost all data series the null hypothesis that the first 20 autocorrelations of the returns as a whole are equal to zero. In the first subperiod only for Boeing and HP this null is not rejected, while in the second subperiod the null is not rejected only for GM, HP, IBM and Walt Disney. Hence HP is the only stock which does not show significant autocorrelation in all periods. When looking at the first to last column with Diebold's (1986) heteroskedasticity-consistent Box-Pierce (1970) Q-statistics it appears that heteroskedasticity indeed affects the inferences about serial correlation in the returns. For the full sample period 1973-2001 and the two subperiods 1973-1986 and 1987-2001 for respectively 18, 9 and 19 data series the null hypothesis of no autocorrelation is not rejected by the adjusted Q-statistic, while it is rejected by the Ljung-Box (1978) Q-statistic. The autocorrelation functions of the squared returns show that for all data series and for all periods the autocorrelations are high and significant up to order 20. The Ljung-Box (1978) Q-statistics reject the null of no autocorrelation in the squared returns firmly. Hence, all data series exhibit significant volatility clustering, that is large (small) shocks are likely to be followed by large (small) shocks.

3.3 Technical trading strategies

We refer to section 2.3 for an overview of the technical trading rules applied in this chapter. In this thesis we mainly confine ourselves to objective trend-following technical trading techniques which can be implemented on a computer. In total we test in this chapter a set of 787 technical trading strategies². This set is divided in three different groups: moving-average rules (in total 425), trading range break-out (also called support-and- resistance) rules (in total 170) and filter rules (in total 192). These strategies are also described by Brock, Lakonishok and LeBaron (1992), Levich and Thomas (1993) and Sullivan, Timmermann and White (1999). We use the parameterizations of Sullivan et al. (1999) as a starting point to construct our sets of trading rules. The parameterizations are presented in Appendix B. If a signal is generated at the end of day t, we assume that the corresponding trading position at day t+1 is executed against the price at the end of day t. Each trading strategy divides the data set of prices in three subsets. A buy (sell) period is defined as the period after a buy (sell) signal up to the next trading signal. A neutral period is defined as the period after a neutral signal up to the next buy or sell signal. The subsets consisting of buy, sell or neutral periods will be called the set of buy sell or neutral days.

3.4 Trading profits

We superimpose the signals of a technical trading rule on the buy-and-hold benchmark. If a buy signal is generated, then money is borrowed against the risk-free interest rate and a double position in the risky asset is held. On a neutral signal only a long position in the risky asset is held, while on a sell signal the position in the risky asset is sold and the proceeds are invested against the risk-free interest rate. If a technical trading rule has forecasting power, then it should beat the buy-and-hold strategy consistently and persistently. It should advise to buy when prices rise and it should advise to sell when prices fall. Therefore its performance, i.e. mean return or Sharpe ratio, will be compared to the buy-and-hold performance to examine whether the trading strategy generates valuable signals. The advantage of this procedure is that it circumvents the question whether it is possible to hold an actual short³ position in an asset. We define P_t as the price of the risky asset, I_t as the investment in the risky asset and S_t as the investment in the risk free asset at the end of period t. The percentage/100 costs of initializing or liquidating a trading position is denoted by c. The real profit during a certain trading position including the costs of initializing and liquidating the trading position is determined as follows:

	I_t-1	S_t-1	I_t	S_t	costs
Initiate a double position
Pos_t-1 � 1 � Pos_t=1	2 P_t-1	-P_t-1	I_t-1+2(P_t-P_t-1)	(1+r^f)S_t-1	c P_t-1
Liquidate a double position
Pos_t=1 � Pos_t+1 � 1			P_t	(1+r^f)S_t-1+P_t	c P_t
Initiate a risk free position
Pos_t-1 � -1 � Pos_t=-1	0	P_t-1	0	(1+r^f)S_t-1	c P_t-1
Liquidate a risk free position
Pos_t=-1 � Pos_t+1 � -1			P_t	(1+r^f)S_t-1-P_t	c P_t
Initiate a long position
Pos_t-1 � 0 � Pos_t=0	P_t-1	0	I_t-1+(P_t-P_t-1)	0	0
Liquidate a long position
Pos_t=0 � Pos_t+1 � 0			P_t	0	0
Position not changed			I_t-1+
Pos_t-1=Pos_t			(1+Pos_t)(P_t-P_t-1)	(1+r^f)S_t-1	0

The profit at day t is equal to (I_t+S_t)-(I_t-1+S_t-1)-costs. The net return of a technical trading strategy during a trading position is then equal to

r_t =

I_t+ S_t-costs

I_t-1 + S_t-1

-1.

Note that because a continuous long position in the risky asset is the benchmark the trading signals are superimposed upon, liquidating the double or risk free position means a return back to the long position. Furthermore, costs are defined to be paid only when a double or risk free position is initialized or liquidated. For example, if a risk free position is held until the end of day t is turned into a double position from the beginning of day t+1, part of the costs, because of liquidating the risk free position at the end of day t, are at the expense of the profit at day t and part of the costs, because of initializing the long position at the beginning of day t+1 against the price at the end of day t, are at the expense of the profit at day t+1. In this chapter, 0, 0.10, 0.25, 0.50, 0.75 and 1% costs per trade are implemented. This wide range of transaction costs captures a range of different trader types. For example, floor traders and large investors, such as mutual funds, can trade against relatively low transaction costs in the range of 0.10 to 0.25%. Home investors face higher costs in the range of 0.25 to 0.75%, depending whether they trade through the internet, by telephone or through their personal account manager. Next, because of the bid-ask spread, extra costs over the transaction costs are faced. By examining a wide range of 0 to 1% costs per trade, we belief that we can capture most of the cost possibilities faced in reality by most of the traders.

3.5 Data snooping

Data snooping is the danger that the performance of the best forecasting model found in a given data set is just the result of chance instead of the result of truly superior forecasting power. The search over many different models should be taken into account before making inferences on the forecasting power of the best model. It is widely acknowledged by empirical researchers that data snooping is a dangerous practice to be avoided. Building on the work of Diebold and Mariano (1995) and West (1996), White (2000) developed a simple and straightforward procedure for testing the null hypothesis that the best model encountered in a specification search has no predictive superiority over a given benchmark model. This procedure is called White's Reality Check (RC) for data snooping. We briefly discuss the method hereafter.

The performance of each technical trading strategy used in this chapter is compared to the benchmark of a buy-and-hold strategy. Predictions are made for M periods, indexed from J+1 through T=J+1+M, where the first J data points are used to initialize the K technical trading strategies, so that each technical trading strategy starts at least generating signals at time t=J+1. The performance of strategy k in excess of the buy-and-hold is defined as f_k. The null hypothesis that the best strategy is not superior to the benchmark of buy-and-hold is given by

H₀:

max

k=1...K

E(f_k) � 0,

where E(.) is the expected value. The alternative hypothesis is that the best strategy is superior to the buy-and-hold benchmark. In this chapter we use two performance/selection criteria. Firstly, we use the mean return of the strategy in excess of the mean return of the buy-and-hold (BH) strategy

_k=

�

t=J+1

r_k,t -

�

t=J+1

r_BH,t =

_k-

_BH.

Secondly, we use the Sharpe ratio of the strategy in excess of the Sharpe ratio of the buy-and-hold strategy in which case

_k=

_k-

s.e.(r_k)

_BH-

s.e.(r_BH)

=Sharpe_k-Sharpe_BH,

where r_f is the mean risk-free interest rate and s.e.(.) is the standard error of the corresponding return series. The Sharpe ratio measures the excess return of a strategy over the risk-free interest rate per unit of risk, as measured by the standard deviation, of the strategy. The higher the Sharpe ratio, the better the reward attained per unit of risk taken.

The null hypothesis can be evaluated by applying the stationary bootstrap algorithm of Politis and Romano (1994). This algorithm resamples blocks with varying length from the original data series, where the block length follows the geometric distribution⁴, to form a bootstrapped data series. The purpose of the stationary bootstrap is to capture and preserve any dependence in the original data series in the bootstrapped data series. The stationary bootstrap algorithm is used to generate B bootstrapped data series. Applying strategy k to the bootstrapped data series yields B bootstrapped values of f_k, denoted as f_k,b^*, where b indexes the bth bootstrapped sample. Finally the RC p-value is determined by comparing the test statistic

max

k=1...K

{M (

_k)} (1)

to the quantiles of

_b^*=

max

k=1...K

{M (

_k,b^*-

_k)}. (2)

In formula this is

�

b=1

_b^*>

)

where 1(.) is an indicator function that takes the value one if and only if the expression within brackets is true. White (2000) applies the Reality Check to a specification search directed toward forecasting the daily returns of the S&P 500 one day in advance in the period May 29, 1988 through May 31, 1994 (the period May 29, 1988 through June 3, 1991 is used as initialization period). In the specification search linear forecasting models that make use of technical indicators, such as momentum, local trend, relative strength indexes and moving averages, are applied to the data set. The mean squared prediction error and directional accuracy are used as prediction measures. White (2000) shows that the Reality Check does not reject the null hypothesis that the best technical indicator model cannot beat the buy-and-hold benchmark. However, if one looks at the p-value of the best strategy not corrected for the specification search, the so called data-mined p-value, the null is not rejected marginally in the case of the mean squared prediction error accuracy, and is rejected in the case of directional accuracy.

Sullivan, Timmermann and White (1999, 2001) utilize the RC to evaluate simple technical trading strategies and calendar effects applied to the Dow-Jones Industrial Average (DJIA) in the period 1897-1996. As performance measures the mean return and the Sharpe ratio are chosen. The benchmark is the buy-and-hold strategy. Sullivan et al. (1999) find for both performance measures that the best technical trading rule has superior forecasting power over the buy-and-hold benchmark in the period 1897-1986 and for several subperiods, while accounting for the effects of data snooping. Thus it is found that the earlier results of Brock et al. (1992) survive the danger of data snooping. However for the period 1986-1996 this result is not repeated. The individual data-mined p-values still reject the null hypothesis, but the RC p-values do not reject the null hypothesis anymore. For the calendar effects (Sullivan et al., 2001) it is found that the individual data-mined p-values do reject the null hypothesis in the period 1897-1996, while the RC, which corrects for the search of the best model, does not reject the null hypothesis of no superior forecasting power of the best model over the buy-and-hold benchmark. Hence Sullivan et al. (1999, 2001) show that if one does not correct for data snooping one can make wrong inferences about the significant forecasting power of the best model.

Hansen (2001) identifies a similarity condition for asymptotic tests of composite hypotheses and shows that this condition is a necessary condition for a test to be unbiased. The similarity condition used is called ``asymptotic similarity on the boundary of a null hypothesis'' and Hansen (2001) shows that White's RC does not satisfy this condition. This causes the RC to be a biased test, which yields inconsistent p-values. Further the RC is sensitive to the inclusion of poor and irrelevant models, because the p-value can be increased by including poor models. The RC is therefore a subjective test, because the null hypothesis can finally be rejected by including enough poor models. Also the RC has unnecessary low power, which can be driven to zero by the inclusion of ``silly'' models. Hansen (2001) concludes that the RC can misguide the researcher to believe that no real forecasting improvement is provided by a class of competing models, even though one of the models indeed is a superior forecasting model. Therefore Hansen (2001) applies within the framework of White (2000) the similarity condition to derive a test for superior predictive ability (SPA), which reduces the influence of poor performing strategies in deriving the critical values. This test is unbiased and is more powerful than the RC. The null hypothesis tested is that none of the alternative models is superior to the benchmark model. The alternative hypothesis is that one or more of the alternative models are superior to the benchmark model. The SPA-test p-value is determined by comparing the test statistic (3.1) to the quantiles of

_b^*=

max

k=1...K

{M (

_k,b^*-g(

_k))}, (3)

where

_k)=

�
�
�
�
�

0, if

_k � -A_k=-

M^-1/4

var(M^1/2

_k )

. (4)

The correction factor A_k depends on an estimate of var(M^1/2 f_k ). A simple estimate can be calculated from the bootstrap resamples as

var(M^1/2

_k )=

�

b=1

(M^1/2

_k,b^*-M^1/2

_k)².

Equations (3.3) and (3.4) ensure that poor and irrelevant strategies cannot have a large impact on the SPA-test p-value, because (3.4) filters the strategy set for these kind of strategies.

Hansen (2001) uses the RC and the SPA-test to evaluate forecasting models applied to US annual inflation in the period 1952 through 2000. The forecasting models are linear regression models with fundamental variables, such as employment, inventory, interest, fuel and food prices, as the regressors. The benchmark model is a random walk and as performance measure the mean absolute deviation is chosen. Hansen (2001) shows that the null hypothesis is neither rejected by the SPA-test p-value, nor by the RC p-value, but that there is a large difference in magnitude between both p-values, likely to be caused by the inclusion of poor models in the space of forecasting models.

3.6 Empirical results

3.6.1 Results for the mean return criterion

Technical trading rule performance

In section 3.2 we have shown that in the subperiod 1973-1986 most stocks could not even beat a risk free investment, while they boosted in the subperiod 1987-2001. However the larger rewards came with greater risks. One may question whether technical trading strategies can persistently generate higher pay-offs than the buy-and-hold benchmark. In total we apply 787 objective computerized trend-following technical trading techniques with and without transaction costs to the DJIA and to the stocks listed in the DJIA. Tables 3.5 and 3.6 show for the full sample period, 1973:1-2001:6, for each data series some statistics of the best strategy selected by the mean return criterion, if 0% and 0.25% costs per trade are implemented. Column 2 shows the parameters of the best strategy. In the case of a moving-average (MA) strategy these parameters are ``[short run MA, long run MA]'' plus the refinement parameters ``[%-band filter, time delay filter, fixed holding period, stop-loss]''. In the case of a trading range break, also called support-and-resistance (SR), strategy, the parameters are ``[the number of days over which the local maximum and minimum is computed]'' plus the refinement parameters as with the moving averages. In the case of a filter (FR) strategy the parameters are ``[the %-filter, time delay filter, fixed holding period]''. Columns 3 and 4 show the mean yearly return and excess mean yearly return of the best-selected strategy over the buy-and-hold benchmark, while columns 5 and 6 show the Sharpe ratio and excess Sharpe ratio of the best strategy over the buy-and-hold benchmark. Column 7 shows the maximum loss the best strategy generates. Columns 8, 9 and 10 show the number of trades, the percentage of profitable trades and the percentage of days profitable trades last. Finally, the last column shows the standard deviation of the returns of the data series during profitable trades divided by the standard deviation of the returns of the data series during non-profitable trades.

To summarize, table 3.7 shows for the full sample period, 1973:1-2001:6, and for the two subperiods, 1973:1-1986:12 and 1987:1-2001:6, for each data series examined, the mean yearly excess return over the buy-and-hold benchmark of the best strategy selected by the mean return criterion, after implementing 0, 0.10, 0.25 and 0.75%⁵ costs per trade.

For transaction costs between 0-1% it is found for each data series that the excess return of the best strategy over the buy-and-hold is positive in almost all cases; the only exception is Caterpillar in the full sample period if 1% costs per trade are implemented. Even for Bethlehem Steel, which stock shows considerable losses in all periods, the best strategy generates not only a positive excess return, but also a positive normal return. By this we mean that the best strategy on its own did generate profits. This is important because excess returns can also be positive in the case when a non-profitable strategy loses less than the buy-and-hold benchmark. If transaction costs increase from 0 to 0.75% per trade, then it can be seen in the last row of table 3.7 that on average the excess return by which the best strategy beats the buy-and-hold benchmark decreases; for example from 19 to 5.34% for the full sample period. Further, the technical trading rules yield the best results in the first subperiod 1973-1986, the period during which the stocks performed the worst. On average, in the case of no transaction costs, the mean excess return in this period is equal to 33% yearly, almost twice as large as in the period 1987-2001, when it is equal to 17.3% yearly. In comparison, the DJIA advanced by 6.1% yearly in the 1973-1986 period, while it advanced by 12.1% yearly in the 1987-2001 period. Thus from these results we can conclude that in all sample periods technical trading rules are capable of beating a buy-and-hold benchmark, also after correction for transaction costs.

From table 3.5 (full sample) it can be seen that in the case of zero transaction costs the best-selected strategies are mainly strategies which generate a lot of trading signals. Trading positions are held for only a few days. For example, the best strategy found for the DJIA is a single crossover moving-average strategy with no extra refinements, which generates a signal when the price series crosses a 2-day moving average. The mean yearly return of this strategy is 25%, which corresponds with a mean yearly excess return of 14.4%. The Sharpe ratio is equal to 0.0438 and the excess Sharpe ratio is equal to 0.0385. The maximum loss of the strategy is 25.1%, while the maximum loss of buying and holding the DJIA is equal to 36.1%. The number of trades executed by following the strategy is very large, once every two days, but also the percentage of profitable trades is very large, namely 69.7%. These profitable trades span 80.8% of the total number of trading days. Although the trading rules show economic significance, they all go through periods of heavy losses, well above the 50% for most stocks (table 3.1). Comparable results are found for the other data series and the two subperiods.

If transaction costs are increased to 0.25% per trade, then table 3.6 shows that the best-selected strategies are strategies which generate substantially fewer signals in comparison with the zero transaction costs case. Trading positions are now held for a longer time. For example, for the DJIA the best strategy generates a trade every 2 years and 4 months. Also the percentage of profitable trades and the percentage of days profitable trades last increases for most data series. Similar results are found in the two subperiods.

CAPM

Dooley and Shafer (1983) notice for floating exchange rates that there is some relationship between variability in the returns, as measured by standard deviation and technical trading rule profits. They find that a large increase in the variability is associated with a dramatic increase in the profitability. If no transaction costs are implemented, then from table 3.5, last column, it can be seen that the standard deviations of the returns of the data series themselves during profitable trades are higher than the standard deviations of the returns during non-profitable trades for almost all stocks, except Exxon Mobil, Home Depot and Wal-Mart Stores. However, if 0.25% costs per trade are implemented, then for 18 data series out of 35 the standard deviation ratio is larger than one. According to the EMH it is not possible to exploit a data set with past information to predict future price changes. The good performance of the technical trading rules could therefore be the reward for holding a risky asset needed to attract investors to bear the risk. Since the technical trading rule forecasts only depend on past price history, it seems unlikely that they should result in unusual risk-adjusted profits. To test this hypothesis we regress Sharpe-Lintner capital asset pricing models (CAPMs)

r_tⁱ-r_t^f=a + b (r_t^DJIA-r_t^f) + e_t. (5)

Here r_tⁱ is the return on day t of the best strategy selected for stock i, r_t^DJIA is the return on day t of the price-weighted Dow-Jones Industrial Average, which represents the market portfolio, and r_t^f is the risk-free interest rate. The coefficient b measures the riskiness of the active technical trading strategy relatively to the passive strategy of buying and holding the market portfolio. If b is not significantly different from one, then it is said that the strategy has equal risk as a buying and holding the market portfolio. If b>1 (b<1), then it is said that the strategy is more risky (less risky) than buying and holding the market portfolio and that it should therefore yield larger (smaller) returns. The coefficient a measures the excess return of the best strategy applied to stock i after correction of bearing risk. If it is not possible to beat a broad market portfolio after correction for risk and hence technical trading rule profits are just the reward for bearing risk, then a should not be significantly different from zero. For the full sample period table 3.8 shows for different transaction cost cases the estimation results, if for each data series the best strategy is selected by the mean return criterion. Estimation is done with Newey-West (1987) heteroskedasticity and autocorrelation consistent (HAC) standard errors. Table 3.9 summarizes the CAPM estimation results for all periods and all transaction cost cases by showing the number of data series for which significant estimates of a or b are found at the 10% significance level.

1973-2001 a<0 a>0 b<1 b>1 a>0 � a>0 �

b<1 b>1

0% 0 29 14 3 11 3

0.10% 0 17 14 3 5 2

0.25% 0 10 13 5 5 1

0.50% 0 7 14 8 3 2

0.75% 0 7 13 13 2 4

1% 0 8 12 13 2 5

1973-1986

0% 0 26 5 6 4 6

0.10% 0 16 7 7 4 3

0.25% 0 9 8 7 1 2

0.50% 0 6 10 6 1 2

0.75% 0 6 12 6 1 2

1% 0 5 12 8 1 3

1987-2001

0% 0 20 19 2 11 2

0.10% 0 11 15 4 3 4

0.25% 0 10 16 3 2 3

0.50% 0 7 16 6 1 4

0.75% 0 7 9 10 1 4

1% 0 7 7 11 1 4

Table 3.9: Summary: significance CAPM estimates, mean return criterion. For all periods and for each transaction cost case, the table shows the number of data series for which significant estimates are found at the 10% significance level for the coefficients in the Sharpe-Lintner CAPM (3.5). Columns 1 and 2 show the number of data series for which the estimate of a is significantly negative and positive. Columns 3 and 4 show the number of data series for which the estimate of b is significantly smaller and larger than one. Column 5 shows the number of data series for which the estimate of a is significantly positive as well as the estimate of b is significantly smaller than one. Column 6 shows the number of data series for which the estimate of a is significantly positive as well as the estimate of b is significantly larger than one. Note that for the periods 1973-2001, 1973-1986 and 1987-2001, the number of data series analyzed is equal to 35, 30 and 35.

For example, for the best strategy applied to the DJIA in the case of zero transaction costs, the estimate of a is significantly positive at the 1% significance level and is equal to 5.39 basis points per day, that is approximately 13.6% per year. The estimate of b is significantly smaller than one at the 10% significance level, which indicates that although the strategy generates a higher reward than simply buying and holding the index, it is less risky. If transaction costs increase, then the estimate of a decreases to 1.91 basis points per day, 4.8% per year, in the case of 1% transaction costs, but is still significantly positive. The estimate of b is significantly smaller than one for all transaction cost cases at the 10% significance level.

As further can be seen in tables 3.8 and 3.9, if no transaction costs are implemented, then for the full sample period the estimate of a is significantly positive for 28 out of 34 stocks. For none of the data series the estimate of a is significantly negative. Thus, for only six stocks the estimate of a is not significantly different from zero. The estimate of a decreases as costs increase and becomes less significant for more data series. In the 0.50% and 1% transaction costs cases, only for respectively 7 and 8 data series out of 35 the estimate of a is significantly positive. Further the estimate of b is significantly smaller than one for 14 data series, if zero transaction costs are implemented. Only for three stocks b is significantly larger than one. Further, table 3.9 shows that for all periods and all transaction cost cases the estimate of a is never significantly negative, indicating that the best strategy is never performing significantly worse than the buy-and-hold benchmark. Also for the two subperiods it is found that for more than half of the data series the estimate of a is significantly positive, if no transaction costs are implemented. Moreover, especially for the second subperiod, it is found that the estimate of b is significantly smaller than one for many data series, indicating that the best strategy is less risky than the market portfolio.

From the findings until now we conclude that there are trend-following technical trading techniques which can profitably be exploited, even after correction for transaction costs, when applied to the DJIA and to the stocks listed in the DJIA in the period 1973-2001 and in the two subperiods 1973-1986 and 1987-2001. As transaction costs increase, the best strategies selected are those which trade less frequently. Furthermore, it becomes more difficult for more and more stocks to reject the null hypothesis that the profit of the best strategy is just the reward of bearing risk. However, for transaction costs up to 1% per trade it is found for a group of stocks that the best strategy, selected by the mean return criterion, can statistically significantly beat the buy-and-hold benchmark strategy. Moreover, for many data series it is found that the best strategy, although it does not necessarily beats the buy-and-hold, is less risky than the buy-and-hold strategy.

Data snooping

The question remains open whether the findings in favour of technical trading for particular stocks are the result of chance or of real superior forecasting power. Therefore we apply White's (2000) Reality Check and Hansen's (2001) Superior Predictive Ability test. Because Hansen (2001) showed that the Reality Check is biased in the direction of one, p-values are computed for both tests to investigate whether these tests lead in some cases to different conclusions.

If the best strategy is selected by the mean return criterion, then table 3.10 shows the nominal, RC and SPA-test p-values for the full sample period 1973-2001 in the case of 0 and 0.10% costs per trade, for the first subperiod 1973-1986 in the case of 0 and 0.25% costs per trade and for the second subperiod 1987-2001 only in the case of 0% costs per trade. Table 3.11 summarizes the results for all periods and all transaction cost cases by showing the number of data series for which the corresponding p-value is smaller than 0.10. That is, the number of data series for which the null hypothesis is rejected at the 10% significance level.

period 1973-2001 1973-1986 1987-2001

costs p_n p_W p_H p_n p_W p_H p_n p_W p_H

0% 35 0 8 30 1 13 34 0 1

0.10% 35 0 0 30 0 3 34 0 0

0.25% 35 0 0 30 0 0 34 0 0

0.50% 35 0 0 30 0 0 34 0 0

0.75% 34 0 0 29 0 0 33 0 0

1% 33 0 0 29 0 0 33 0 0

Table 3.11: Summary: Testing for predictive ability, mean return criterion. For all periods and for each transaction cost case, the table shows the number of data series for which the nominal (p_n), White's (2000) Reality Check (p_W) or Hansen's (2001) Superior Predictive Ability test (p_H) p-value is smaller than 0.10. Note that for the periods 1973-2001, 1973-1986 and 1987-2001, the number of data series analyzed is equal to 35, 30 and 35.

The nominal p-value, also called data mined p-value, tests the null hypothesis that the best strategy is not superior to the buy-and-hold benchmark, but does not correct for data snooping. From the tables it can be seen that this null hypothesis is rejected for all periods and for all cost cases at the 10% significance level. However, for the full sample period, if we correct for data snooping, then we find, in the case of no transaction costs, that for all of the data series the null hypothesis that the best strategy is not superior to the benchmark after correcting for data snooping is not rejected by the RC. However, for 8 data series the null hypothesis that none of the strategies are superior to the benchmark after correcting for data snooping is rejected by the SPA-test. In 8 cases the two data snooping tests lead thus to different inferences about predictive ability of technical trading in the 1973-2001 period. For these 8 cases the biased RC misguides by not rejecting the null, even though one of the technical trading strategies is indeed superior, as shown by the SPA-test. However, if we implement as little as 0.10% costs for the full sample period, then both tests do not reject their null anymore for all data series.

For the subperiod 1973-1986 we find that the SPA-test p-value does reject the null for 13 data series, while the RC p-value does reject the null for only 1 data series at the 10% significance level. However, if 0.25% costs are implemented, then both tests do not reject their null for all data series. For the second subperiod 1987-2001 we find that the two tests are in agreement. Even if no transaction costs are implemented, then both tests do not reject the null at the 10% significance level in almost all cases. Hence, we conclude that the best strategy, selected by the mean return criterion, is not capable of beating the buy-and-hold benchmark strategy, after a correction is made for transaction costs and data snooping.

3.6.2 Results for the Sharpe ratio criterion

Technical trading rule performance

Similar to tables 3.5 and 3.6, table 3.12 shows for the full sample period for some data series some statistics of the best strategy selected by the Sharpe ratio criterion, if 0 or 0.25% costs per trade are implemented. Only the results for those data series are presented for which the best strategy selected by the Sharpe ratio criterion differs from the best strategy selected by the mean return criterion. To summarize, table 3.13 shows for all periods and for each data series the Sharpe ratio of the best strategy selected by the Sharpe ratio criterion, after implementing 0, 0.10, 0.25 or 0.75% costs per trade, in excess of the Sharpe ratio of the buy-and-hold benchmark. It is found that the Sharpe ratio of the best-selected strategy in excess of the Sharpe ratio of the buy-and-hold is positive in almost all cases; the only exceptions are Caterpillar in the full sample period and Wal-Mart Stores in the last subperiod, both in the case of 1% transaction costs. If transaction costs increase from 0 to 0.75%, then in the last row of table 3.13 it can be seen that for the full sample period the excess Sharpe ratio declines on average from 0.0258 to 0.0078. For the full sample period table 3.12 shows that the best strategies selected in the case of zero transaction costs are mainly strategies that generate a lot of signals. Trading positions are held for only a short period. Moreover, for most data series the best-selected strategy is the same as in the case that the best strategy is selected by the mean return criterion. If costs are increased to 0.25% per trade, then the best-selected strategies generate fewer signals and trading positions are held for longer periods. Now for 14 data series the best-selected strategy differs from the case when the best strategy is selected by the mean return criterion. For the two subperiods similar results are found. However the excess Sharpe ratios are higher in the period 1973-1986 than in the period 1987-2001.

As for the mean return criterion it is found that for each data series the best strategy, selected by the Sharpe ratio criterion, beats the buy-and-hold benchmark and that this strategy can profitably be exploited, even after correction for transaction costs. The results show that technical trading strategies were most profitable in the period 1973-1986, but also profits are made in the period 1987-2001.

CAPM

The estimation results of the Sharpe-Lintner CAPM in tables 3.14 and 3.15 for the Sharpe ratio selection criterion are similar to the estimation results in tables 3.8 and 3.9 for the mean return selection criterion. In the case of zero transaction costs for most data series the estimate of a is significantly positive, but as costs increase, then we find for fewer data series a significantly positive estimate of a.

1973-2001 a<0 a>0 b<1 b>1 a>0 � a>0 �

b<1 b>1

0% 0 29 14 1 11 1

0.10% 0 18 19 2 8 1

0.25% 0 13 18 4 6 2

0.50% 0 9 17 6 4 3

0.75% 0 9 14 9 4 4

1% 0 9 14 10 3 4

1973-1986

0% 0 26 5 6 4 6

0.10% 0 15 8 5 3 3

0.25% 0 8 10 3 0 1

0.50% 0 6 10 4 0 1

0.75% 0 5 11 4 0 1

1% 0 4 12 7 0 1

1987-2001

0% 0 25 21 1 16 1

0.10% 0 16 20 0 7 0

0.25% 0 11 19 1 3 1

0.50% 0 7 19 2 2 2

0.75% 0 7 18 2 2 2

1% 0 7 12 3 1 3

Table 3.15: Summary: significance CAPM estimates, Sharpe ratio criterion. For all periods and for each transaction cost case, the table shows the number of data series for which significant estimates are found at the 10% significance level for the coefficients in the Sharpe-Lintner CAPM (3.5). Columns 1 and 2 show the number of data series for which the estimate of a is significantly negative and positive. Columns 3 and 4 show the number of data series for which the estimate of b is significantly smaller and larger than one. Column 5 shows the number of data series for which the estimate of a is significantly positive as well as the estimate of b is significantly smaller than one. Column 6 shows the number of data series for which the estimate of a is significantly positive as well as the estimate of b is significantly larger than one. Note that for the periods 1973-2001, 1973-1986 and 1987-2001, the number of data series analyzed is equal to 35, 30 and 35.

Data snooping

If the best strategy is selected by the Sharpe ratio criterion, then table 3.16 shows the nominal, White's RC and Hansen's SPA-test p-values for all periods and different transaction costs cases. The results are shown for the full sample period 1973-2001 in the case of 0 and 0.10% costs per trade and for the two subperiods 1973-1986 and 1987-2001 in the case of 0 and 0.25% costs per trade. Table 3.17 summarizes the results for all periods and all transaction cost cases by showing the number of data series for which the corresponding p-value is smaller than 0.10.

period 1973-2001 1973-1986 1987-2001

costs p_n p_W p_H p_n p_W p_H p_n p_W p_H

0% 35 4 16 30 10 21 35 0 5

0.10% 35 0 3 30 0 5 35 0 2

0.25% 35 0 0 30 0 0 35 0 1

0.50% 34 0 0 30 0 0 35 0 1

0.75% 30 0 0 30 0 0 34 0 1

1% 29 0 0 28 0 0 34 0 1

Table 3.17: Summary: Testing for predictive ability, Sharpe ratio criterion. For all periods and for each transaction cost case, the table shows the number of data series for which the nominal (p_n), White's (2000) Reality Check (p_W) or Hansen's (2001) Superior Predictive Ability test (p_H) p-value is smaller than 0.10. Note that for the periods 1973-2001, 1973-1986 and 1987-2001, the number of data series analyzed is equal to 35, 30 and 35.

If the nominal p-value is used to test the null hypothesis that the best strategy is not superior to the buy-and-hold benchmark, then the null is rejected in all periods for most data series at the 5% significance level. For the full sample period, if a correction is made for data snooping, then it is found, in the case of zero transaction costs, that for 4 data series the null hypothesis that the best strategy is not superior to the benchmark after correcting for data snooping is rejected by the RC at the 10% significance level. However, for 16 data series the null hypothesis that none of the strategies is superior to the benchmark after correcting for data snooping is rejected by the SPA-test. Thus for 12 data series the RC leads to wrong inferences about the forecasting power of the best-selected strategy. However, if we implement as little as 0.10% costs, then these contradictory results only occur for 3 data series (the null is rejected for none of the data series by the RC) and if we increase the costs even further to 0.25%, then for none of the data series either test rejects the null. In the first subperiod 1973-1986, if zero transaction costs are implemented, then the RC p-value rejects the null for 10 data series, while the SPA-test p-value rejects the null for 21 data series. For the second subperiod 1987-2001, if no transaction costs are implemented, then the results of both tests are more in conjunction. The RC rejects the null for none of the data series, while the SPA-test rejects the null for 5 data series. If transaction costs are increased to 0.25%, then for both subperiods both tests do not reject the null for almost all data series. Only for Goodyear Tire, in the last subperiod, the SPA-test does reject the null, even in the 1% costs case. Hence, we conclude that the best strategy, selected by the Sharpe ratio criterion, is not capable of beating the benchmark of a buy-and-hold strategy, after a correction is made for transaction costs and data snooping.

3.7 A recursive out-of-sample forecasting approach

Like most academic literature on technical analysis, we investigated the profitability and forecastability of technical trading rules in sample, instead of out of sample. White's (2000) RC and Hansen's (2001) SPA-test, as we applied them, are indeed in-sample test procedures as they test whether the best strategy in a certain trading period has significant forecasting power, after correction for the search for the best strategy in that specific trading period. However, whether a technical trading strategy applied to a financial time series in a certain period shows economically/statistically significant forecasting power does not say much about its future performance. If it shows forecasting power, then profits earned in the past do not necessarily imply that profits can also be made in the future. On the other hand, if the strategy does not show forecasting power, then it could be that during certain subperiods the strategy was actually performing very well due to some characteristics in the data, but the same strategy was loosing during other subperiods, because the characteristics of the data changed. Therefore, only inferences about the forecastability of technical analysis can be made by testing whether strategies that performed well in the past, also perform well in the future. In this section we test the forecasting power of our set of trend-following technical trading techniques by applying a recursive optimizing and testing procedure. For example, recursively at the beginning of each month we investigate which technical trading rule performed the best in the preceding six months (training period) and we select this best strategy to generate trading signals during the coming month (testing period). Sullivan et al. (1999) also apply a recursive out-of-sample forecasting procedure. However, in their procedure, the strategy which performed the best from t=0 is selected to make one step ahead forecasts. We instead use a moving window, as in Lee and Mathur (1995), in which strategies are compared and the best strategy is selected to make forecasts for some period thereafter.

Our approach is similar to the recursive modeling, estimation and forecasting approach of Pesaran and Timmermann (1995, 2000) and Marquering and Verbeek (2000). They use a collection of macro-economic variables as information set to base trading decisions upon. A linear regression model, with a subset of the macro-economic variables as regressors and the excess return of the risky asset over the risk-free interest rate as dependent variable, is estimated recursively with ordinary least squares. The subset of macro-economic variables which yields the best fit to the excess returns is selected to make an out-of-sample forecast of the excess return for the next period. According to a certain trading strategy a position in the market is chosen on the basis of the forecast. They show that historical fundamental information can help in predicting excess returns. We will do essentially the same for technical trading strategies, using only past observations from the financial time series itself.

We define the training period on day t to last from t-Tr until and including t-1, where Tr is the length of the training period. The testing period lasts from t until and including t+Te-1, where Te is the length of the testing period. For each of the 787 strategies the performance during the training period is computed. Then the best technical trading strategy is selected by the mean return or Sharpe ratio criterion and is applied in the testing period to generate trading signals. After the end of the testing period this procedure is repeated again until the end of the data series is reached. For the training and testing periods we use 36 different parameterizations of [Tr, Te] which can be found in Appendix C.

In the case of 0.25% transaction costs tables 3.18 and 3.19 show for the DJIA and for each stock in the DJIA some statistics of the best recursive optimizing and testing procedure, if the best strategy in the training period is selected by the mean return and Sharpe ratio criterion respectively. Because the longest training period is five years, the results are computed for the period 1978:10-2001:6. Table 3.20A, B (i.e. table 3.20 panel A, panel B) summarizes the results for both selection criteria in the case of 0, 0.10 and 0.50% costs per trade. In the second to last row of table 3.20A it can be seen that, if in the training period the best strategy is selected by the mean return criterion, then the excess return over the buy-and-hold of the best recursive optimizing and testing procedure is, on average, 12.3, 6.9, 2.8 and -1.2% yearly in the case of 0, 0.10, 0.25 and 0.50% costs per trade. Thus the excess returns decline on average sharply when implementing as little as 0.10% costs. If the Sharpe ratio criterion is used for selecting the best strategy during the training period, then the Sharpe ratio of the best recursive optimizing and testing procedure in excess of the Sharpe ratio of the buy-and-hold benchmark is on average 0.0145, 0.0077, 0.0031 and -0.0020 in the case of 0, 0.10, 0.25 and 0.50% costs per trade, also declining sharply when low costs are implemented (see second to last row of table 3.20B). Thus in our recursive out-of-sample testing procedures small transaction costs cause forecastability to disappear.

For comparison, the last row in table 3.20A, B shows the average over the results of the best strategies selected by the mean return or Sharpe ratio criterion in sample for each data series tabulated. As can be seen, clearly the results of the best strategies selected in sample are better than the results of the best recursive out-of-sample forecasting procedure.

If the mean return selection criterion is used, then table 3.21A shows for the 0 and 0.10% transaction cost cases⁶ for each data series the estimation results of the Sharpe-Lintner CAPM (see equation 3.5) where the return of the best recursive optimizing and testing procedure in excess of the risk-free interest rate is regressed against a constant a and the return of the DJIA in excess of the risk-free interest rate. Estimation is done with Newey-West (1987) heteroskedasticity and autocorrelation consistent (HAC) standard errors. Table 3.22 summarizes the CAPM estimation results for all transaction cost cases by showing the number of data series for which significant estimates of a and b are found at the 10% significance level. In the case of zero transaction costs for 12 data series out of 35 the estimate of a is significantly positive at the 10% significance level. This number decreases to 3 (1, 0) if 0.10% (0.25, 0.50%) costs per trade are implemented. Table 3.21B shows the results of the CAPM estimation for the case that the best strategy in the training period is selected by the Sharpe ratio criterion. Now in the case of zero transaction costs for 14 data series it is found that the estimate of a is significantly positive at the 10% significance level. If transaction costs increase to 0.10% (0.25, 0.50%), then only for 7 (6, 1) out of 35 data series the estimate of a is significantly positive. Hence, after correction for transaction costs and risk it can be concluded, independently of the selection criterion used, that the best recursive optimizing and testing procedure shows no statistically significant out-of-sample forecasting power.

Selection criterion: mean return

costs a<0 a>0 b<1 b>1 a>0 � a>0 �

b<1 b>1

0% 0 12 13 3 5 2

0.10% 0 3 12 5 2 1

0.25% 0 1 8 7 0 1

0.50% 1 0 7 7 0 0

Selection criterion: Sharpe ratio

costs a<0 a>0 b<1 b>1 a>0 � a>0 �

b<1 b>1

0% 0 14 15 4 7 1

0.10% 0 7 16 3 2 0

0.25% 1 6 14 3 2 1

0.50% 0 1 12 4 0 0

Table 3.22: Summary: significance CAPM estimates for best out-of-sample testing procedure. For each transaction cost case, the table shows the number of data series for which significant estimates are found at the 10% significance level for the coefficients in the Sharpe-Lintner CAPM. Columns 1 and 2 show the number of data series for which the estimate of a is significantly negative and positive. Columns 3 and 4 show the number of data series for which the estimate of b is significantly smaller and larger than one. Column 5 shows the number of data series for which the estimate of a is significantly positive as well as the estimate of b is significantly smaller than one. Column 6 shows the number of data series for which the estimate of a is significantly positive as well as the estimate of b is significantly larger than one.

3.8 Conclusion

In this chapter we apply a set of 787 objective computerized trend-following technical trading techniques to the Dow-Jones Industrial Average (DJIA) and to 34 stocks listed in the DJIA in the period January 1973 through June 2001. For each data series the best technical trading strategy is selected by the mean return or Sharpe ratio criterion. Because numerous research papers found that technical trading rules show some forecasting power in the era until 1987, but not in the period thereafter, we split our sample in two subperiods: 1973-1986 and 1987-2001. We find for all periods and for both selection criteria that for each data series a technical trading strategy can be selected that is capable of beating the buy-and-hold benchmark, even after correction for transaction costs. Although buy-and-hold stock investments had difficulty in beating a continuous risk free investment during the 1973-1986 subsample, the strongest results in favour of technical trading are found for this subperiod. For example, in the full sample period 1973-2001 it is found that the best strategy beats the buy-and-hold benchmark on average with 19, 10, 7.5, 6.1, 5.3 and 4.9% yearly in the case of 0, 0.10, 0.25, 0.50, 0.75 and 1% transaction costs, if the best strategy is selected by the mean return criterion. These are quite substantial numbers.

The profits generated by the technical trading strategies could be the reward necessary to attract investors to bear the risk of holding the asset. To test this hypothesis we estimate Sharpe-Lintner CAPMs. For each data series the daily return of the best strategy in excess of the risk-free interest rate is regressed against a constant (a) and the daily return of the DJIA in excess of the risk-free interest rate. The coefficient of the last regression term is called b and measures the riskiness of the strategy relatively to buying and holding the market portfolio. If technical trading rules do not generate excess profits after correction for risk, then a should not be significantly different from zero. If no transaction costs are implemented, then we find for both selection criteria that in all periods for most data series the estimate of a is significantly positive. This means that the best selected technical trading rules show forecasting power after a correction is made for risk. However, if costs are increased, we are less able to reject the null hypothesis that technical trading rule profits are the reward for bearing risk. But still, in numerous cases the estimate of a is significantly positive.

An important question is whether the positive results found in favour of technical trading are due to chance or the fact that the best strategy has genuine superior forecasting power over the buy-and-hold benchmark. This is called the danger of data snooping. We apply White's (2000) Reality Check (RC) and Hansen's (2001) Superior Predictive Ability (SPA) test, to test the null hypothesis that the best strategy found in a specification search is not superior to the benchmark of a buy-and-hold if a correction is made for data snooping. Hansen (2001) showed that the RC is sensitive to the inclusion of poor and irrelevant forecasting rules. Because we compute p-values for both tests, we can investigate whether both test procedures lead to contradictory inferences. If no transaction costs are implemented, then we find for the mean return and the Sharpe ratio criterion that the RC and the SPA-test in some cases lead to different conclusions, especially for the subperiod 1973-1986. The SPA-test finds in numerous cases that the best strategy does beat the buy-and-hold significantly after correction for data snooping and the implementation of bad strategies. Thus the biased RC misguides the researcher in several cases by not rejecting the null. However, if as little as 0.25% costs per trade are implemented, then both tests lead for both selection criteria, for all sample periods and for all data series to the same conclusion: the best strategy is not capable of beating the buy-and-hold benchmark after a correction is made for the specification search that is used to find the best strategy. We therefore finally conclude that the good performance of trend-following technical trading techniques applied to the DJIA and to the individual stocks listed in the DJIA, especially in the 1973-1986 subperiod, is merely the result of chance than of good forecasting power.

Next we apply a recursive optimizing and testing method to test whether the best strategy found in a specification search during a training period shows also forecasting power during a testing period thereafter. For example, every month the best strategy from the last 6 months is selected to generate trading signals during that month. In total we examine 36 different training and testing period combinations. In the case of no transaction costs, the best recursive optimizing and testing procedure yields on average an excess return over the buy-and-hold of 12.3% yearly, if the best strategy in the training period is selected by the mean return criterion. Thus the best strategy found in the past continues to generate good results in the future. However, if as little as 0.25% transaction costs are implemented, then the excess return decreases to 2.8%. Finally, estimation of Sharpe-Lintner CAPMs shows that, after correction for transaction costs and risk, the best recursive optimizing and testing procedure has no statistically significant forecasting power anymore.

Hence, in short, after correcting for transaction costs, risk, data snooping and out-of-sample forecasting, we conclude that objective trend-following technical trading techniques applied to the DJIA and to the stocks listed in the DJIA in the period 1973-2001 are not genuine superior, as suggested by their performances, to the buy-and-hold benchmark.

Appendix

A. Tables

Table
3.1	Data series examined, sample and maximum cumulative loss
3.2	Summary statistics: 1973-2001
3.3	Summary statistics: 1973-1986
3.4	Summary statistics: 1987-2001
3.5	Statistics best strategy: 1973-2001, mean return criterion, 0% costs
3.6	Statistics best strategy: 1973-2001, mean return criterion, 0.25% costs
3.7	Mean return best strategy in excess of mean return buy-and-hold
3.8	Estimation results CAPM: 1973-2001, mean return criterion
3.9	Summary: significance CAPM estimates, mean return criterion
3.10	Testing for predictive ability: mean return criterion
3.11	Summary: Testing for predictive ability, mean return criterion
3.12	Statistics best strategy: 1973-2001, Sharpe ratio criterion, 0 and 0.25% costs
3.13	Sharpe ratio best strategy in excess of Sharpe ratio buy-and-hold
3.14	Estimation results CAPM: 1973-2001, Sharpe ratio criterion
3.15	Summary: significance CAPM estimates, Sharpe ratio criterion
3.16	Testing for predictive ability: Sharpe ratio criterion
3.17	Summary: Testing for predictive ability, Sharpe Ratio criterion
3.18	Statistics best out-of-sample testing procedure: mean return criterion, 0.25% costs
3.19	Statistics best out-of-sample testing procedure: Sharpe ratio criterion, 0.25% costs
3.20A	Mean return best out-of-sample testing procedure in excess of mean return buy-and-hold
3.20B	Sharpe ratio best out-of-sample testing procedure in excess of Sharpe ratio buy-and-hold
3.21A	Estimation results CAPM for best out-of-sample testing procedure: mean return criterion
3.21B	Estimation results CAPM for best out-of-sample testing procedure: Sharpe ratio criterion
3.22	Summary: significance CAPM estimates for best out-of-sample testing procedure

Table 3.1: Data series examined, sample period and largest cumulative loss. Column 1 shows the names of 34 stocks listed in the DJIA in the period 1973:1-2001:6. Column 2 shows their respective sample periods. Columns 3 and 4 show the largest cumulative loss of the data series in %/100 terms and the period during which this decline occurred.

Data set Sample period Max. loss Period of max. loss

DJIA 12/31/73 - 06/29/01 -0.3613 08/26/87 - 10/19/87

ALCOA 12/31/73 - 06/29/01 -0.4954 04/17/74 - 12/05/74

AMERICAN EXPRESS 12/31/73 - 06/29/01 -0.6313 03/07/74 - 10/03/74

AT&T 12/31/73 - 06/29/01 -0.7326 02/04/99 - 12/28/00

BETHLEHEM STEEL 12/31/73 - 06/29/01 -0.9655 03/11/76 - 12/06/00

BOEING 12/31/73 - 06/29/01 -0.6632 01/31/80 - 06/28/82

CATERPILLAR 12/31/73 - 06/29/01 -0.6064 04/27/81 - 12/13/84

CHEVRON - TEXACO 12/31/73 - 06/29/01 -0.5823 11/27/80 - 08/04/82

CITIGROUP 10/27/87 - 06/29/01 -0.5652 04/07/98 - 10/07/98

COCA - COLA 12/31/73 - 06/29/01 -0.6346 01/04/74 - 10/03/74

E.I. DU PONT DE NEMOURS 12/31/73 - 06/29/01 -0.5347 05/21/98 - 09/26/00

EASTMAN KODAK 12/31/73 - 06/29/01 -0.6551 04/02/76 - 03/06/78

EXXON MOBIL 12/31/73 - 06/29/01 -0.4448 01/04/74 - 10/03/74

GENERAL ELECTRIC 12/31/73 - 06/29/01 -0.5333 01/07/74 - 09/13/74

GENERAL MOTORS 12/31/73 - 06/29/01 -0.5652 01/03/77 - 02/22/82

GOODYEAR TIRE 12/31/73 - 06/29/01 -0.8165 08/12/87 - 11/09/90

HEWLETT - PACKARD 12/31/73 - 06/29/01 -0.6579 10/06/87 - 11/07/90

HOME DEPOT 12/28/84 - 06/29/01 -0.5385 08/12/87 - 10/26/87

HONEYWELL INTL. 09/17/86 - 06/29/01 -0.5123 06/22/99 - 06/27/00

INTEL 12/28/79 - 06/29/01 -0.6978 09/01/00 - 04/04/01

INTL. BUS. MACH. 12/31/73 - 06/29/01 -0.7654 08/21/87 - 08/16/93

INTERNATIONAL PAPER 12/31/73 - 06/29/01 -0.6073 03/11/76 - 04/17/80

J.P. MORGAN CHASE & CO. 12/31/73 - 06/29/01 -0.8165 03/28/86 - 10/31/90

JOHNSON & JOHNSON 12/31/73 - 06/29/01 -0.4758 06/10/74 - 04/25/77

MCDONALDS 12/31/73 - 06/29/01 -0.6667 06/11/74 - 10/04/74

MERCK 12/31/73 - 06/29/01 -0.4957 01/06/92 - 04/15/94

MICROSOFT 03/11/87 - 06/29/01 -0.6516 12/28/99 - 12/20/00

MINNESOTA MNG. & MNFG. 12/31/73 - 06/29/01 -0.4546 01/04/74 - 04/04/78

PHILIP MORRIS 12/31/73 - 06/29/01 -0.6759 11/24/98 - 02/16/00

PROCTER & GAMBLE 12/31/73 - 06/29/01 -0.5445 01/12/00 - 03/10/00

SBC COMMUNICATIONS 11/16/84 - 06/29/01 -0.4045 07/19/99 - 02/23/00

SEARS, ROEBUCK & CO. 12/31/73 - 06/29/01 -0.6746 06/10/74 - 12/11/80

UNITED TECHNOLOGIES 12/31/73 - 06/29/01 -0.5099 01/07/81 - 03/17/82

WAL - MART STORES 12/30/81 - 06/29/01 -0.5047 08/24/87 - 12/03/87

WALT DISNEY 12/31/73 - 06/29/01 -0.6667 03/14/74 - 12/16/74

Table 3.2: Summary statistics: 1973-2001. The first column shows the names of the the stocks listed in the DJIA in the period 1973:1-2001:6. Columns 2 to 7 show the number of observations, the mean yearly effective return in %/100 terms, the mean, standard deviation, skewness and kurtosis of the daily logarithmic return. Column 8 shows the t-ratio testing whether the mean daily return is significantly different from zero. Column 9 shows the Sharpe ratio. Column 10 shows the largest cumulative loss in %/100 terms. Column 11 shows the Ljung-Box (1978) Q-statistic testing whether the first 20 autocorrelations of the return series as a whole are significantly different from zero. Column 12 shows the heteroskedasticity adjusted Box-Pierce (1970) Q-statistic, as derived by Diebold (1986). The final column shows the Ljung-Box (1978) Q-statistic for testing autocorrelations in the squared returns. Significance level of the (adjusted) Q(20)-test statistic can be evaluated based on the following chi-squared values: a) chi-squared(0.99,20)=37.57, b) chi-squared(0.95,20)=31.41, c) chi-squared(0.90,20)=28.41.

Data set N Yearly Mean Std.Dev. Skew. Kurt. t-ratio Sharpe Max.loss Q20 Adj Q20 Q20 r2

DJIA 7174 0.0923 0.00035 0.010181 -2.305 63.561 2.91a 0.005243 -0.3613 49.00a 13.01 362.23a

ALCOA 7174 0.0943 0.000358 0.019203 -0.298 12.348 1.58 0.003158 -0.4954 67.11a 30.84c 635.10a

AMERICAN EXPRESS 7174 0.088 0.000335 0.02133 -0.381 11.968 1.33 0.001763 -0.6313 64.77a 30.16c 1650.62a

AT&T 7174 0.0487 0.000189 0.016008 -0.341 17.641 1 -0.006762 -0.7326 43.64a 17.7 1472.98a

BETHLEHEM STEEL 7174 -0.0935 -0.000389 0.027128 -0.309 18.861 -1.22 -0.025299 -0.9655 55.58a 27.76 774.12a

BOEING 7174 0.1882 0.000684 0.019737 -0.036 7.126 2.94a 0.019631 -0.6632 47.48a 37.15b 311.57a

CATERPILLAR 7174 0.0541 0.000209 0.018176 -0.462 12.07 0.97 -0.004834 -0.6064 83.27a 42.37a 722.22a

CHEVRON - TEXACO 7174 0.0855 0.000326 0.016315 -0.058 7.771 1.69c 0.001761 -0.5823 71.30a 48.26a 901.65a

CITIGROUP 3568 0.2742 0.000962 0.021851 0.12 6.162 2.63a 0.033446 -0.5652 39.53a 28.66c 385.97a

COCA - COLA 7174 0.1055 0.000398 0.016902 -0.481 19.144 1.99b 0.005978 -0.6346 67.97a 27.55 1509.56a

E.I. DU PONT DE NEMOURS 7174 0.0615 0.000237 0.016621 -0.207 8.485 1.21 -0.003624 -0.5347 34.18b 24.82 532.20a

EASTMAN KODAK 7174 0.0048 0.000019 0.018006 -1.435 40.483 0.09 -0.015443 -0.6551 42.88a 24.03 460.21a

EXXON MOBIL 7174 0.0994 0.000376 0.01396 -0.672 26.309 2.28b 0.005675 -0.4448 115.49a 45.74a 1359.07a

GENERAL ELECTRIC 7174 0.1355 0.000504 0.015753 -0.188 8.987 2.71a 0.013153 -0.5333 43.32a 21.33 2287.65a

GENERAL MOTORS 7174 0.0471 0.000183 0.017382 -0.204 10.017 0.89 -0.006567 -0.5652 31.61b 22.46 782.11a

GOODYEAR TIRE 7174 0.0467 0.000181 0.018877 -0.593 20.93 0.81 -0.006129 -0.8165 77.94a 38.60a 522.31a

HEWLETT - PACKARD 7174 0.1254 0.000469 0.022922 -0.148 8.719 1.73c 0.0075 -0.6579 23.93 16.28 539.19a

HOME DEPOT 4305 0.3251 0.001117 0.023751 -0.917 19.57 3.09a 0.036954 -0.5385 54.39a 34.95b 122.29a

HONEYWELL INTL. 3857 0.0846 0.000322 0.021072 -0.554 39.883 0.95 0.004262 -0.5123 57.91a 24.53 639.75a

INTEL 5610 0.2199 0.000789 0.028326 -0.249 7.909 2.09b 0.017476 -0.6978 60.98a 37.65a 1090.84a

INTL. BUS. MACH. 7174 0.0725 0.000278 0.01719 -0.347 17.36 1.37 -0.001123 -0.7654 32.99b 18.31 344.98a

INTERNATIONAL PAPER 7174 0.0361 0.000141 0.018256 -0.491 18.143 0.65 -0.008552 -0.6073 61.30a 36.78b 475.13a

J.P. MORGAN CHASE & CO. 7174 0.0721 0.000276 0.019916 -0.468 16.47 1.17 -0.001042 -0.8165 39.65a 20.75 344.59a

JOHNSON & JOHNSON 7174 0.1134 0.000426 0.015839 -0.241 9.105 2.28b 0.008162 -0.4758 87.46a 50.82a 1046.42a

MCDONALDS 7174 0.1094 0.000412 0.01771 -0.432 11.923 1.97b 0.006488 -0.6667 48.44a 25 1384.84a

MERCK 7174 0.1249 0.000467 0.015916 -0.01 6.44 2.49b 0.010692 -0.4957 56.64a 36.36b 930.26a

MICROSOFT 3732 0.3845 0.001291 0.024908 -0.936 18.997 3.17a 0.042494 -0.6516 50.17a 22.16 499.16a

MINNESOTA MNG. & MNFG. 7174 0.0655 0.000252 0.014968 -0.998 28.906 1.42 -0.003025 -0.4546 41.29a 24.22 325.92a

PHILIP MORRIS 7174 0.1406 0.000522 0.017649 -0.537 15.258 2.50b 0.01275 -0.6759 62.53a 42.50a 197.11a

Table 3.2 continued.

Data set N Yearly Mean Std.Dev. Skew. Kurt. t-ratio Sharpe Max.loss Q20 Adj Q20 Q20 r2

PROCTER & GAMBLE 7174 0.0882 0.000335 0.01567 -2.846 77.504 1.81c 0.002459 -0.5445 65.86a 26.65 193.97a

SBC COMMUNICATIONS 4335 0.1231 0.000461 0.016595 -0.596 21.051 1.83c 0.013299 -0.4045 46.22a 27.58 965.78a

SEARS, ROEBUCK & CO. 7174 0.0371 0.000145 0.018911 -0.161 15.76 0.65 -0.008048 -0.6746 65.60a 27.41 692.74a

UNITED TECHNOLOGIES 7174 0.1469 0.000544 0.016836 -0.075 6.358 2.74a 0.01467 -0.5099 61.04a 41.26a 731.90a

WAL - MART STORES 5087 0.2808 0.000982 0.020218 -0.011 5.566 3.46a 0.03542 -0.5047 55.92a 37.72a 1030.70a

WALT DISNEY 7174 0.1296 0.000484 0.020449 -0.927 21.334 2.00b 0.009125 -0.6667 35.35b 11.95 571.60a

Table 3.3: Summary statistics: 1973-1986. The first column shows the names of the the stocks listed in the DJIA in the period 1973:1-1986:12. Columns 2 to 7 show the number of observations, the mean yearly effective return in %/100 terms, the mean, standard deviation, skewness and kurtosis of the daily logarithmic return. Column 8 shows the t-ratio testing whether the mean daily return is significantly different from zero. Column 9 shows the Sharpe ratio. Column 10 shows the largest cumulative loss in %/100 terms. Column 11 shows the Ljung-Box (1978) Q-statistic testing whether the first 20 autocorrelations of the return series as a whole are significantly different from zero. Column 12 shows the heteroskedasticity adjusted Box-Pierce (1970) Q-statistic, as derived by Diebold (1986). The final column shows the Ljung-Box (1978) Q-statistic for testing autocorrelations in the squared returns. Significance level of the (adjusted) Q(20)-test statistic can be evaluated based on the following chi-squared values: a) chi-squared(0.99,20)=37.57, b) chi-squared(0.95,20)=31.41, c) chi-squared(0.90,20)=28.41.

Data set N Yearly Mean Std.Dev. Skew. Kurt. t-ratio Sharpe Max.loss Q20 Adj Q20 Q20 r2

DJIA 3392 0.0613 0.000236 0.009268 0.266 4.718 1.48 -0.014284 -0.3522 64.77a 47.24a 807.23a

ALCOA 3392 0.0251 0.000098 0.018056 -0.06 6.484 0.32 -0.014967 -0.4954 68.61a 46.06a 347.63a

AMERICAN EXPRESS 3392 0.0709 0.000272 0.020474 0.05 5.654 0.77 -0.004728 -0.6313 74.87a 44.34a 1468.10a

AT&T 3392 0.0529 0.000205 0.011221 0.164 6.987 1.06 -0.014608 -0.3076 32.24b 23.36 282.74a

BETHLEHEM STEEL 3392 -0.1163 -0.000491 0.021015 -0.752 17.242 -1.36 -0.040883 -0.8995 76.24a 37.35b 181.61a

BOEING 3392 0.2799 0.000979 0.020775 0.145 4.478 2.75a 0.029395 -0.6632 19.18 16.08 224.74a

CATERPILLAR 3392 -0.008 -0.000032 0.015946 -0.233 8.102 -0.12 -0.025105 -0.6064 97.70a 72.98a 141.18a

CHEVRON - TEXACO 3392 0.0734 0.000281 0.017345 0.289 5.114 0.94 -0.005055 -0.5823 71.72a 53.56a 326.42a

COCA - COLA 3392 0.0441 0.000171 0.015861 -0.063 7.04 0.63 -0.012442 -0.6346 75.94a 41.54a 1223.19a

E.I. DU PONT DE NEMOURS 3392 0.0348 0.000136 0.015226 0.153 5.243 0.52 -0.015284 -0.5171 33.40b 27.21 212.82a

EASTMAN KODAK 3392 -0.0088 -0.000035 0.016197 0.296 6.154 -0.13 -0.024932 -0.6551 38.17a 28.15 631.40a

EXXON MOBIL 3392 0.0846 0.000322 0.012691 0.029 4.334 1.48 -0.003655 -0.4448 50.18a 40.90a 228.64a

GENERAL ELECTRIC 3392 0.0775 0.000296 0.014956 0.135 5.481 1.15 -0.004828 -0.5333 36.78b 21.05 2111.06a

GENERAL MOTORS 3392 0.0302 0.000118 0.015072 0.171 5.658 0.46 -0.016623 -0.5652 39.94a 29.52c 275.71a

GOODYEAR TIRE 3392 0.0779 0.000298 0.016501 0.36 5.811 1.05 -0.004301 -0.5648 29.00c 21.68 311.47a

HEWLETT - PACKARD 3392 0.1113 0.000419 0.019543 0.058 4.91 1.25 0.002576 -0.4795 27.12 22.75 212.75a

INTEL 1828 0.032 0.000125 0.027913 0.068 3.85 0.19 -0.010579 -0.6277 42.82a 37.00b 66.58a

INTL. BUS. MACH. 3392 0.0507 0.000196 0.013597 0.456 5.724 0.84 -0.012678 -0.3937 29.04c 21.23 342.40a

Table 3.3 continued.

Data set N Yearly Mean Std.Dev. Skew. Kurt. t-ratio Sharpe Max.loss Q20 Adj Q20 Q20 r2

INTERNATIONAL PAPER 3392 0.0277 0.000108 0.016462 0.246 5.347 0.38 -0.015803 -0.6073 76.39a 61.62a 129.18a

J.P. MORGAN CHASE & CO. 3392 0.0635 0.000244 0.016143 0.173 4.846 0.88 -0.007706 -0.4443 32.71b 23.76 389.22a

JOHNSON & JOHNSON 3392 0.0422 0.000164 0.015091 0.093 5.144 0.63 -0.013552 -0.4758 62.21a 40.44a 896.24a

MCDONALDS 3392 0.0997 0.000377 0.018283 -0.521 14.002 1.2 0.000475 -0.6667 66.60a 28.67c 896.54a

MERCK 3392 0.0869 0.000331 0.014654 0.163 6.012 1.31 -0.002578 -0.437 57.45a 34.04b 804.60a

MINNESOTA MNG. & MNFG. 3392 0.0304 0.000119 0.013763 0.305 4.901 0.5 -0.018158 -0.4546 85.93a 61.15a 678.87a

PHILIP MORRIS 3392 0.1269 0.000474 0.015716 0.107 5.412 1.76c 0.006706 -0.4206 67.12a 48.22a 541.40a

PROCTER & GAMBLE 3392 0.0384 0.00015 0.012195 0.173 5.09 0.71 -0.017961 -0.4021 57.27a 39.11a 863.04a

SEARS, ROEBUCK & CO. 3392 -0.0007 -0.000003 0.016463 0.314 5.266 -0.01 -0.022557 -0.6746 56.78a 39.07a 513.68a

UNITED TECHNOLOGIES 3392 0.1645 0.000604 0.016661 0.125 4.444 2.11b 0.014156 -0.5099 33.99b 28.56c 132.37a

WAL - MART STORES 1305 0.5225 0.001668 0.020035 0.107 4.286 3.01a 0.065144 -0.3191 29.04c 24.22 97.21a

WALT DISNEY 3392 0.1083 0.000408 0.020927 -0.596 10.832 1.14 0.001877 -0.6667 39.13a 24.61 214.93a

Table 3.4: Summary statistics: 1987-2001. The first column shows the names of the stocks listed in the DJIA in the period 1987:1-2001:6. Columns 2 to 7 show the number of observations, the mean yearly effective return in %/100 terms, the mean, standard deviation, skewness and kurtosis of the daily logarithmic return. Column 8 shows the t-ratio testing whether the mean daily return is significantly different from zero. Column 9 shows the Sharpe ratio. Column 10 shows the largest cumulative loss in %/100 terms. Column 11 shows the Ljung-Box (1978) Q-statistic testing whether the first 20 autocorrelations of the return series as a whole are significantly different from zero. Column 12 shows the heteroskedasticity adjusted Box-Pierce (1970) Q-statistic, as derived by Diebold (1986). The final column shows the Ljung-Box (1978) Q-statistic for testing autocorrelations in the squared returns. Significance level of the (adjusted) Q(20)-test statistic can be evaluated based on the following chi-squared values: a) chi-squared(0.99,20)=37.57, b) chi-squared(0.95,20)=31.41, c) chi-squared(0.90,20)=28.41.

Data set N Yearly Mean Std.Dev. Skew. Kurt. t-ratio Sharpe Max.loss Q20 Adj Q20 Q20 r2

DJIA 3781 0.1209 0.000453 0.010937 -3.681 88.53 2.55b 0.020124 -0.3613 44.74a 10.26 186.47a

ALCOA 3781 0.1604 0.00059 0.02018 -0.456 15.507 1.80c 0.017719 -0.4673 44.13a 22.9 342.83a

AMERICAN EXPRESS 3781 0.1035 0.000391 0.022074 -0.688 16.051 1.09 0.007169 -0.5652 48.82a 19.23 830.77a

AT&T 3781 0.0449 0.000174 0.019322 -0.395 15.058 0.55 -0.003016 -0.7326 31.81b 15.5 653.99a

BETHLEHEM STEEL 3781 -0.0725 -0.000299 0.031628 -0.177 16.356 -0.58 -0.016801 -0.9415 32.68b 19.57 393.34a

BOEING 3781 0.1116 0.00042 0.018759 -0.266 10.499 1.38 0.009988 -0.4853 47.40a 36.79b 139.73a

CATERPILLAR 3781 0.1131 0.000425 0.019968 -0.567 12.788 1.31 0.00964 -0.5397 30.80c 16.33 435.70a

CHEVRON - TEXACO 3781 0.0966 0.000366 0.015336 -0.505 11.389 1.47 0.008688 -0.4308 47.23a 30.39c 645.95a

CITIGROUP 3568 0.2742 0.000962 0.021851 0.12 6.162 2.63a 0.033446 -0.5652 39.53a 28.66c 385.97a

COCA - COLA 3781 0.1637 0.000601 0.017785 -0.75 25.666 2.08b 0.020736 -0.5127 64.68a 23.98 818.69a

E.I. DU PONT DE NEMOURS 3781 0.0859 0.000327 0.017783 -0.41 9.769 1.13 0.005317 -0.5347 36.44b 23.85 282.15a

EASTMAN KODAK 3781 0.0171 0.000067 0.019491 -2.301 53.344 0.21 -0.008479 -0.6134 30.89c 16.75 243.15a

Table 3.4 continued.

Data set N Yearly Mean Std.Dev. Skew. Kurt. t-ratio Sharpe Max.loss Q20 Adj Q20 Q20 r2

EXXON MOBIL 3781 0.1129 0.000425 0.01501 -1.044 35.377 1.74c 0.01279 -0.3312 134.02a 39.32a 748.31a

GENERAL ELECTRIC 3781 0.1901 0.000691 0.016437 -0.411 11.039 2.58a 0.027863 -0.4117 41.85a 17.67 1042.40a

GENERAL MOTORS 3781 0.0626 0.000241 0.019223 -0.364 10.794 0.77 0.000429 -0.4787 27.99 19.4 424.53a

GOODYEAR TIRE 3781 0.0196 0.000077 0.020783 -0.999 24.937 0.23 -0.007497 -0.8165 71.95a 34.53b 262.80a

HEWLETT - PACKARD 3781 0.1382 0.000514 0.025581 -0.227 9.167 1.23 0.010988 -0.6579 24.62 16.7 221.20a

HOME DEPOT 3781 0.3754 0.001265 0.022779 -1.232 22.137 3.41a 0.04531 -0.5385 68.00a 38.76a 139.68a

HONEYWELL INTL. 3781 0.0868 0.00033 0.021218 -0.553 39.576 0.96 0.00461 -0.5123 57.90a 24.54 626.18a

INTEL 3781 0.3228 0.00111 0.028526 -0.394 9.723 2.39b 0.030756 -0.6978 68.01a 35.49b 889.02a

INTL. BUS. MACH. 3781 0.0924 0.000351 0.019872 -0.563 17.332 1.09 0.005943 -0.7654 24.87 15.83 131.51a

INTERNATIONAL PAPER 3781 0.0437 0.00017 0.019731 -0.867 22.912 0.53 -0.003181 -0.5527 38.80a 23.89 260.81a

J.P. MORGAN CHASE & CO. 3781 0.0799 0.000305 0.02278 -0.651 17.168 0.82 0.003173 -0.7952 28.93c 16.11 132.03a

JOHNSON & JOHNSON 3781 0.1814 0.000661 0.016483 -0.477 11.521 2.47b 0.026017 -0.3838 86.63a 51.54a 497.17a

MCDONALDS 3781 0.1181 0.000443 0.017184 -0.334 9.425 1.58 0.012237 -0.4833 34.80b 21.07 550.76a

MERCK 3781 0.1602 0.000589 0.016972 -0.115 6.46 2.14b 0.021025 -0.4957 46.11a 32.96b 319.04a

MICROSOFT 3732 0.3845 0.001291 0.024908 -0.936 18.997 3.17a 0.042494 -0.6516 50.17a 22.16 499.16a

MINNESOTA MNG. & MNFG. 3781 0.098 0.000371 0.015976 -1.738 39.909 1.43 0.00866 -0.3736 45.38a 21.57 161.36a

PHILIP MORRIS 3781 0.1531 0.000565 0.019223 -0.844 18.412 1.81c 0.017296 -0.6759 48.37a 33.24b 70.86a

PROCTER & GAMBLE 3781 0.1349 0.000502 0.018234 -3.49 79.433 1.69c 0.014789 -0.5445 64.75a 26.14 95.27a

SBC COMMUNICATIONS 3781 0.1018 0.000385 0.017332 -0.567 20.129 1.37 0.008779 -0.4045 44.97a 27.55 833.06a

SEARS, ROEBUCK & CO. 3781 0.0723 0.000277 0.020868 -0.373 18.355 0.82 0.00213 -0.6202 56.42a 21.86 344.34a

UNITED TECHNOLOGIES 3781 0.1314 0.00049 0.016995 -0.243 7.932 1.77c 0.015126 -0.5 49.71a 30.84c 535.93a

WAL - MART STORES 3781 0.2067 0.000746 0.02028 -0.049 5.979 2.26b 0.025299 -0.5047 55.25a 34.79b 938.94a

WALT DISNEY 3781 0.1491 0.000552 0.020017 -1.263 32.498 1.69c 0.01593 -0.4454 23.83 5.3 366.16a

Table 3.5: Statistics best strategy: 1973-2001, mean return criterion, 0% costs. Statistics of the best strategy, selected by the mean return criterion, if no costs are implemented, for each data series listed in the first column. Column 2 shows the strategy parameters. Columns 3 and 4 show the mean return and excess mean return on a yearly basis in %/100 terms. Columns 5 and 6 show the Sharpe and excess Sharpe ratio. Column 7 shows the largest cumulative loss of the strategy in %/100 terms. Columns 8, 9 and 10 show the number of trades, the percentage of profitable trades and the percentage of days these profitable trades lasted. The last column shows the standard deviation of returns during profitable trades divided by the standard deviation of returns during non-profitable trades.

Data set Strategy parameters r r^e S S^e ML # tr. %tr.>0 %d > 0 SDR

DJIA [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.2491 0.1435 0.0438 0.0385 -0.2507 3531 0.697 0.808 1.3909

ALCOA [ MA: 1, 2, 0.001, 0, 0, 0.000] 0.5131 0.3827 0.0506 0.0474 -0.6360 2885 0.693 0.811 1.2083

AMERICAN EXPRESS [ MA: 1, 2, 0.001, 0, 0, 0.000] 0.3262 0.2190 0.0287 0.0270 -0.8875 3100 0.679 0.793 1.3362

Table 3.5 continued.

Data set Strategy parameters r r^e S S^e ML # tr. %tr.>0 %d > 0 SDR

AT&T [ FR: 0.020, 0, 50 ] 0.1225 0.0704 0.0080 0.0148 -0.6056 232 0.681 0.776 1.2719

BETHLEHEM STEEL [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.2104 0.3352 0.0136 0.0389 -0.9926 3247 0.649 0.804 1.2600

BOEING [ MA: 1, 2, 0.001, 0, 0, 0.000] 0.4936 0.2570 0.0473 0.0277 -0.7850 2895 0.697 0.808 1.2081

CATERPILLAR [ MA: 1, 2, 0.001, 0, 0, 0.000] 0.4904 0.4139 0.0521 0.0569 -0.7815 2895 0.690 0.814 1.1647

CHEVRON - TEXACO [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.2746 0.1741 0.0308 0.0290 -0.7196 3466 0.674 0.797 1.3414

CITIGROUP [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.5930 0.2502 0.0540 0.0206 -0.6954 1707 0.689 0.805 1.3224

COCA - COLA [ MA: 1, 2, 0.001, 0, 0, 0.000] 0.2841 0.1616 0.0314 0.0254 -0.6220 2925 0.682 0.793 1.3365

E.I. DU PONT DE NEMOURS [ MA: 1, 2, 0.001, 0, 0, 0.000] 0.2079 0.1379 0.0201 0.0238 -0.5995 3049 0.682 0.800 1.2779

EASTMAN KODAK [ MA: 2, 5, 0.000, 0, 25, 0.000] 0.0776 0.0725 0.0000 0.0154 -0.7019 440 0.634 0.720 1.1390

EXXON MOBIL [ MA: 1, 2, 0.000, 0, 5, 0.000] 0.1754 0.0691 0.0163 0.0106 -0.6480 1472 0.653 0.762 0.8588

GENERAL ELECTRIC [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.2388 0.0910 0.0263 0.0132 -0.5838 3417 0.674 0.788 1.3597

GENERAL MOTORS [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.1652 0.1127 0.0132 0.0198 -0.8473 3466 0.670 0.795 1.2530

GOODYEAR TIRE [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.2413 0.1859 0.0219 0.0280 -0.8510 3383 0.669 0.794 1.2870

HEWLETT - PACKARD [ MA: 10, 100, 0.000, 0, 50, 0.000] 0.2321 0.0948 0.0211 0.0136 -0.6425 134 0.724 0.704 1.0201

HOME DEPOT [ FR: 0.010, 0, 5, ] 0.6428 0.2398 0.0487 0.0118 -0.8918 904 0.667 0.733 0.8617

HONEYWELL INTL. [ MA: 1, 50, 0.000, 0, 50, 0.000] 0.2714 0.1722 0.0332 0.0289 -0.4040 106 0.717 0.824 1.3056

INTEL [ MA: 1, 2, 0.001, 0, 0, 0.000] 0.4092 0.1552 0.0278 0.0103 -0.8327 2166 0.696 0.801 1.1638

INTL. BUS. MACH. [ MA: 10, 25, 0.000, 2, 0, 0.000] 0.1727 0.0934 0.0155 0.0166 -0.7380 287 0.714 0.826 1.1015

INTERNATIONAL PAPER [ MA: 1, 2, 0.001, 0, 0, 0.000] 0.2937 0.2486 0.0293 0.0378 -0.9370 3021 0.688 0.808 1.3174

J.P. MORGAN CHASE & CO. [ MA: 1, 2, 0.001, 0, 0, 0.000] 0.4459 0.3487 0.0434 0.0444 -0.7045 2907 0.695 0.812 1.3051

JOHNSON & JOHNSON [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.3304 0.1949 0.0397 0.0315 -0.3393 3407 0.687 0.796 1.3344

MCDONALDS [ MA: 1, 2, 0.001, 0, 0, 0.000] 0.2676 0.1427 0.0258 0.0193 -0.7242 2950 0.680 0.792 1.1585

MERCK [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.3409 0.1920 0.0379 0.0272 -0.6204 3369 0.688 0.786 1.2630

MICROSOFT [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.7089 0.2343 0.0590 0.0165 -0.6995 1786 0.680 0.807 1.3912

MINNESOTA MNG. & MNFG. [ FR: 0.005, 0, 10 ] 0.2485 0.1718 0.0300 0.0330 -0.4865 895 0.683 0.753 1.1536

PHILIP MORRIS [ MA: 1, 2, 0.001, 0, 0, 0.000] 0.2434 0.0901 0.0227 0.0099 -0.8874 2927 0.693 0.810 1.0819

PROCTER & GAMBLE [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.2340 0.1339 0.0261 0.0236 -0.5526 3455 0.683 0.785 1.4065

SBC COMMUNICATIONS [ MA: 2, 10, 0.000, 0, 25, 0.000] 0.2837 0.1430 0.0376 0.0243 -0.4012 249 0.743 0.806 1.0756

SEARS, ROEBUCK & CO. [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.2411 0.1966 0.0224 0.0305 -0.6622 3474 0.664 0.804 1.3426

UNITED TECHNOLOGIES [ MA: 1, 2, 0.001, 0, 0, 0.000] 0.4957 0.3042 0.0561 0.0414 -0.4957 2942 0.697 0.809 1.3119

WAL - MART STORES [ MA: 1, 2, 0.000, 0, 5, 0.000] 0.5028 0.1733 0.0498 0.0144 -0.6527 1055 0.667 0.748 0.9853

WALT DISNEY [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.3978 0.2374 0.0379 0.0288 -0.7506 3329 0.679 0.807 1.2720

Table 3.6: Statistics best strategy: 1973-2001, mean return criterion, 0.25% costs. Statistics of the best strategy, selected by the mean return criterion, if 0.25% costs per trade are implemented, for each data series listed in the first column. Column 2 shows the strategy parameters. Columns 3 and 4 show the mean return and excess mean return on a yearly basis in %/100 terms. Columns 5 and 6 show the Sharpe and excess Sharpe ratio. Column 7 shows the largest cumulative loss of the strategy in %/100 terms. Columns 8, 9 and 10 show the number of trades, the percentage of profitable trades and the percentage of days these profitable trades lasted. The last column shows the standard deviation of returns during profitable trades divided by the standard deviation of returns during non-profitable trades.

Data set Strategy parameters r r^e S S^e ML # tr. %tr.>0 %d>0 SDR

DJIA [ MA: 5, 25, 0.050, 0, 0, 0.000] 0.1480 0.0511 0.0205 0.0153 -0.4619 12 0.833 0.990 0.7946

ALCOA [ MA: 2, 5, 0.050, 0, 0, 0.000] 0.1513 0.0522 0.0094 0.0063 -0.6804 17 0.647 0.990 0.4365

AMERICAN EXPRESS [ FR: 0.020, 0, 50 ] 0.1790 0.0838 0.0130 0.0113 -0.6908 237 0.679 0.822 0.9709

AT&T [ SR: 150, 0.025, 0, 0, 0.000 ] 0.1087 0.0573 0.0055 0.0123 -0.5060 9 0.889 0.975 0.6723

BETHLEHEM STEEL [ FR: 0.050, 0, 25 ] 0.0797 0.1911 0.0002 0.0255 -0.9433 376 0.604 0.670 0.9242

BOEING [ FR: 0.090, 3, 0 ] 0.2441 0.0471 0.0197 0.0001 -0.7031 174 0.575 0.781 0.8738

CATERPILLAR [ SR: 5, 0.000, 0, 25, 0.000 ] 0.1109 0.0540 0.0044 0.0093 -0.8183 415 0.593 0.654 0.9245

CHEVRON - TEXACO [ FR: 0.020, 0, 25 ] 0.1685 0.0766 0.0146 0.0129 -0.7379 426 0.648 0.722 1.0155

CITIGROUP [ FR: 0.200, 3, 0 ] 0.5070 0.1829 0.0470 0.0136 -0.3511 13 0.846 0.989 0.6553

COCA - COLA [ SR: 250, 0.025, 0, 0, 0.000 ] 0.1765 0.0643 0.0210 0.0150 -0.5210 6 0.833 0.988 1.1012

E.I. DU PONT DE NEMOURS [ MA: 1, 2, 0.000, 0, 10, 0.000] 0.1003 0.0367 0.0038 0.0075 -0.5472 821 0.404 0.485 1.0763

EASTMAN KODAK [ MA: 10, 100, 0.000, 0, 50, 0.000] 0.0462 0.0414 -0.0058 0.0096 -0.7573 139 0.669 0.641 0.9110

EXXON MOBIL [ SR: 200, 0.000, 4, 0, 0.000 ] 0.1512 0.0472 0.0128 0.0072 -0.4821 15 0.867 0.941 1.3361

GENERAL ELECTRIC [ MA: 5, 25, 0.050, 0, 0, 0.000] 0.1884 0.0467 0.0172 0.0041 -0.5417 33 0.727 0.879 1.0058

GENERAL MOTORS [ SR: 10, 0.050, 0, 0, 0.000 ] 0.1443 0.0929 0.0114 0.0179 -0.7442 12 0.917 0.991 0.6222

GOODYEAR TIRE [ SR: 100, 0.050, 0, 0, 0.000 ] 0.1519 0.1006 0.0104 0.0166 -0.8253 6 0.833 0.996 0.8818

HEWLETT - PACKARD [ MA: 10, 100, 0.000, 0, 50, 0.000] 0.2180 0.0824 0.0193 0.0118 -0.6585 134 0.724 0.704 1.0201

HOME DEPOT [ FR: 0.010, 0, 50 ] 0.4603 0.1023 0.0374 0.0004 -0.5940 129 0.752 0.855 1.0079

HONEYWELL INTL. [ MA: 1, 50, 0.000, 0, 50, 0.000] 0.2499 0.1526 0.0300 0.0258 -0.4248 106 0.708 0.821 1.2001

INTEL [ MA: 2, 5, 0.000, 0, 50, 0.000] 0.2772 0.0471 0.0166 -0.0009 -0.8191 187 0.615 0.777 0.8705

INTL. BUS. MACH. [ SR: 150, 0.025, 0, 0, 0.000 ] 0.1199 0.0443 0.0074 0.0085 -0.7342 9 0.778 0.844 1.5333

INTERNATIONAL PAPER [ FR: 0.080, 0, 25 ] 0.0831 0.0455 0.0010 0.0095 -0.5070 293 0.618 0.652 1.0820

J.P. MORGAN CHASE & CO. [ FR: 0.010, 0, 50 ] 0.2050 0.1241 0.0149 0.0159 -0.8588 218 0.656 0.813 1.0024

JOHNSON & JOHNSON [ MA: 2, 5, 0.000, 0, 50, 0.000] 0.1674 0.0486 0.0150 0.0069 -0.6904 228 0.711 0.854 1.0281

MCDONALDS [ FR: 0.140, 0, 50 ] 0.1655 0.0507 0.0173 0.0109 -0.4554 93 0.710 0.743 1.0633

MERCK [ MA: 25, 50, 0.050, 0, 0, 0.000] 0.1847 0.0532 0.0183 0.0076 -0.5552 20 0.750 0.876 0.9966

MICROSOFT [ FR: 0.090, 4, 0 ] 0.5263 0.1026 0.0422 -0.0003 -0.8167 108 0.472 0.761 0.9004

MINNESOTA MNG. & MNFG. [ FR: 0.005, 0, 10 ] 0.1443 0.0741 0.0122 0.0153 -0.5572 895 0.439 0.417 1.0209

PHILIP MORRIS [ FR: 0.300, 4, 0 ] 0.2017 0.0537 0.0168 0.0041 -0.7543 9 0.889 0.989 1.0253

PROCTER & GAMBLE [ MA: 1, 10, 0.000, 0, 50, 0.000] 0.1374 0.0453 0.0088 0.0063 -0.7707 229 0.699 0.795 0.9380

SBC COMMUNICATIONS [ MA: 2, 10, 0.000, 0, 25, 0.000] 0.2367 0.1013 0.0301 0.0169 -0.4101 249 0.675 0.708 1.0608

table 3.6 continued.

Data set Strategy parameters r r^e S S^e ML # tr. %tr.>0 %d > 0 SDR

SEARS, ROEBUCK & CO. [ SR: 150, 0.050, 0, 0, 0.000 ] 0.1307 0.0903 0.0086 0.0166 -0.7156 7 0.857 0.979 1.3855

UNITED TECHNOLOGIES [ SR: 25, 0.000, 0, 50, 0.000 ] 0.2136 0.0582 0.0261 0.0115 -0.3739 178 0.680 0.792 1.0157

WAL - MART STORES [ FR: 0.180, 4, 0 ] 0.3694 0.0693 0.0344 -0.0010 -0.6722 32 0.688 0.906 0.9330

WALT DISNEY [ FR: 0.500, 2, 0 ] 0.1831 0.0475 0.0138 0.0047 -0.7408 2 1.000 1.000 NA

Table 3.7: Mean return best strategy in excess of mean return buy-and-hold. Mean return of the best strategy, selected by the mean return criterion after implementing 0, 0.10, 0.25 and 0.75% costs per trade, in excess of the mean return of the buy-and-hold benchmark, for the full sample period 1973:1-2001:6 and the two subperiods 1973:1-1986:12 and 1987:1-2001:6, for each data series listed in the first column.

period 1973:1-2001:6 1973:1-1986:12 1987:1-2001:6

Data set 0% 0.10% 0.25% 0.75% 0% 0.10% 0.25% 0.75% 0% 0.10% 0.25% 0.75%

DJIA 0.1435 0.0521 0.0511 0.0474 0.1841 0.0883 0.0873 0.0841 0.1255 0.0649 0.0423 0.0298

ALCOA 0.3827 0.1819 0.0522 0.0463 0.5183 0.3403 0.1546 0.0610 0.2757 0.2122 0.1588 0.0391

AMERICAN EXPRESS 0.2190 0.0979 0.0838 0.0441 0.6626 0.4124 0.1906 0.1049 0.1678 0.1501 0.1240 0.1024

AT&T 0.0704 0.0613 0.0573 0.0546 0.1135 0.0651 0.0651 0.0649 0.1047 0.0952 0.0927 0.0845

BETHLEHEM STEEL 0.3352 0.2148 0.1911 0.1152 0.5176 0.3088 0.2135 0.2093 0.2686 0.2545 0.2335 0.1658

BOEING 0.2570 0.0667 0.0471 0.0315 0.4157 0.2253 0.1121 0.1104 0.2032 0.0963 0.0747 0.0287

CATERPILLAR 0.4139 0.2164 0.0540 0.0033 0.6032 0.3787 0.1630 0.0451 0.2633 0.1989 0.1714 0.0840

CHEVRON - TEXACO 0.1741 0.1023 0.0766 0.0276 0.4633 0.2247 0.1549 0.0685 0.1075 0.0988 0.0859 0.0504

CITIGROUP 0.2502 0.1858 0.1829 0.1734 0.2502 0.1858 0.1829 0.1734

COCA - COLA 0.1616 0.0690 0.0643 0.0628 0.2896 0.1422 0.1137 0.1128 0.0874 0.0712 0.0667 0.0655

E.I. DU PONT DE NEMOURS 0.1379 0.0914 0.0367 0.0259 0.2720 0.1473 0.0898 0.0816 0.2225 0.1483 0.0783 0.0387

EASTMAN KODAK 0.0725 0.0552 0.0414 0.0315 0.1639 0.1591 0.1519 0.1283 0.1485 0.1097 0.0743 0.0137

EXXON MOBIL 0.0691 0.0486 0.0472 0.0423 0.1990 0.0954 0.0938 0.0888 0.0751 0.0676 0.0563 0.0355

GENERAL ELECTRIC 0.0910 0.0506 0.0467 0.0369 0.2029 0.1025 0.0898 0.0484 0.0817 0.0788 0.0754 0.0653

GENERAL MOTORS 0.1127 0.0940 0.0929 0.0892 0.2836 0.1510 0.1503 0.1479 0.1721 0.1668 0.1589 0.1329

GOODYEAR TIRE 0.1859 0.1010 0.1006 0.0992 0.2340 0.2143 0.1854 0.0935 0.2220 0.2108 0.1967 0.1939

HEWLETT - PACKARD 0.0948 0.0898 0.0824 0.0578 0.1572 0.0975 0.0851 0.0484 0.1585 0.1298 0.1228 0.0994

HOME DEPOT 0.2398 0.1703 0.1023 0.0870 0.2966 0.1557 0.1249 0.1191

HONEYWELL INTL. 0.1722 0.1643 0.1526 0.1141 0.1639 0.1567 0.1459 0.1105

INTEL 0.1552 0.0608 0.0471 0.0336 0.3187 0.2515 0.2253 0.1746 0.1187 0.1119 0.1063 0.0877

INTL. BUS. MACH. 0.0934 0.0718 0.0443 0.0416 0.0904 0.0821 0.0747 0.0625 0.1541 0.1364 0.1104 0.1017

INTERNATIONAL PAPER 0.2486 0.0817 0.0455 0.0133 0.5821 0.3493 0.0996 0.0366 0.1601 0.1277 0.0980 0.0526

J.P. MORGAN CHASE & CO. 0.3487 0.1513 0.1241 0.0808 0.3981 0.2144 0.1375 0.0960 0.3723 0.1539 0.1453 0.1170

JOHNSON & JOHNSON 0.1949 0.0619 0.0486 0.0207 0.3234 0.1401 0.1258 0.1036 0.0859 0.0832 0.0790 0.0654

Table 3.7 continued.

period 1973:1-2001:6 1973:1-1986:12 1987:1-2001:6

Data set 0% 0.10% 0.25% 0.75% 0% 0.10% 0.25% 0.75% 0% 0.10% 0.25% 0.75%

MCDONALDS 0.1427 0.0557 0.0507 0.0346 0.3379 0.2068 0.1097 0.0850 0.0892 0.0735 0.0626 0.0499

MERCK 0.1920 0.0759 0.0532 0.0465 0.3521 0.1595 0.0539 0.0517 0.1239 0.0911 0.0770 0.0689

MICROSOFT 0.2343 0.1269 0.1026 0.0643 0.2343 0.1269 0.1026 0.0643

MINNESOTA MNG. & MNFG. 0.1718 0.1317 0.0741 0.0028 0.3252 0.1381 0.1245 0.0803 0.2104 0.1692 0.1099 0.0225

PHILIP MORRIS 0.0901 0.0606 0.0537 0.0507 0.2130 0.1416 0.1288 0.0869 0.1168 0.1118 0.1051 0.0838

PROCTER & GAMBLE 0.1339 0.0587 0.0453 0.0255 0.1804 0.0844 0.0614 0.0581 0.1026 0.0679 0.0558 0.0424

SBC COMMUNICATIONS 0.1430 0.1261 0.1013 0.0464 0.1369 0.1201 0.0953 0.0325

SEARS, ROEBUCK & CO. 0.1966 0.0908 0.0903 0.0885 0.3897 0.1639 0.1219 0.1178 0.1344 0.1261 0.1155 0.0892

UNITED TECHNOLOGIES 0.3042 0.1060 0.0582 0.0282 0.3033 0.1084 0.0988 0.0675 0.3050 0.1043 0.0847 0.0323

WAL - MART STORES 0.1733 0.1024 0.0693 0.0543 0.4780 0.3904 0.2686 0.1880 0.1747 0.1026 0.0546 0.0346

WALT DISNEY 0.2374 0.0560 0.0475 0.0474 0.3354 0.1353 0.0885 0.0883 0.1558 0.1044 0.0974 0.0744

Average 0.1898 0.1009 0.0748 0.0534 0.3303 0.1906 0.1273 0.0932 0.1734 0.1275 0.1076 0.0758

Table 3.8: Estimation results CAPM: 1973-2001, mean return criterion. Coefficient estimates of the Sharpe-Lintner CAPM in the period January 1, 1973 through June 29, 2001: r_tⁱ-r_t^f=a + b (r_t^DJIA-r_t^f) + e_t. That is, the excess return of the best strategy, selected by the mean return criterion, over the risk-free interest rate is regressed against a constant and the excess return of the DJIA over the risk-free interest rate. Estimation results for the 0, 0.10, 0.25, 0.50 and 0.75% costs per trade cases are shown. a, b, c indicates that the corresponding coefficient is, in the case of a, significantly different from zero, or in the case of b, significantly different from one, at the 1, 5, 10% significance level. Estimation is done with Newey-West (1987) heteroskedasticity and autocorrelation consistent (HAC) standard errors.

costs per trade 0% 0.10% 0.25% 0.50% 0.75%

Data set a b a b a b a b a b

DJIA 0.000539a 0.872c 0.000210b 0.837 0.000206b 0.837c 0.000199b 0.838c 0.000191c 0.839c

ALCOA 0.001299a 0.887 0.000677b 0.885 0.000199 1.206a 0.000188 1.207a 0.000177 1.207a

AMERICAN EXPRESS 0.000762a 1.150c 0.000352 1.06 0.000301 1.06 0.000215 1.061 0.000144 1.228a

AT&T 0.000125 0.681a 9.18E-05 0.681a 6.59E-05 0.87 6.05E-05 0.87 5.51E-05 0.87

BETHLEHEM STEEL 0.000417 0.829c 3.94E-05 0.869 -3.90E-05 0.868 -0.00017 0.867 -0.000301 0.867

BOEING 0.001248a 0.879 0.000597b 0.878 0.000516 1.017 0.000448 1.271a 0.000442 1.271a

CATERPILLAR 0.001245a 0.781b 0.000648b 0.773a 5.32E-05 1.264 -9.66E-05 1.263 -0.000125 0.942c

CHEVRON - TEXACO 0.000630a 0.667a 0.000376 0.744a 0.000282 0.744a 0.000187 0.672a 0.000101 0.672a

CITIGROUP 0.001320a 1.160b 0.000971a 1.702a 0.000962a 1.701a 0.000947a 1.701a 0.000931a 1.701a

COCA - COLA 0.000651a 0.832b 0.000315 0.955 0.000307c 0.782a 0.000304c 0.783a 0.000301c 0.783a

E.I. DU PONT DE NEMOURS 0.000406c 0.872 0.000238 0.918 3.34E-05 0.918 -1.70E-05 1.108 -1.79E-05 1.108

EASTMAN KODAK -4.80E-05 0.918 -0.000114 0.918 -0.000169 0.977 -0.000222 1.431c -0.000232 1.431c

Table 3.8 continued.

costs per trade 0% 0.10% 0.25% 0.50% 0.75%

Data set a b a b a b a b a b

EXXON MOBIL 0.00029 1.027 0.000217 0.956 0.000211 0.956 0.000202 0.956 0.000193 0.956

GENERAL ELECTRIC 0.000499b 1.017 0.000345c 1.085 0.000324 1.213 0.000302 1.214 0.000266 1.611a

GENERAL MOTORS 0.000264 0.861 0.000206 0.674a 0.000202 0.674a 0.000195 0.674a 0.000188 0.674a

GOODYEAR TIRE 0.000519c 0.789a 0.000234 0.603a 0.000232 0.604a 0.000229 0.605a 0.000226 0.606a

HEWLETT - PACKARD 0.000474c 1.083 0.000455c 1.083 0.000428c 1.083 0.000382 1.082 0.000336 1.082

HOME DEPOT 0.001311a 1.578c 0.001082b 1.578c 0.001009b 0.969 0.000927b 1.102 0.000815b 1.460a

HONEYWELL INTL. 0.000581c 0.611a 0.000554c 0.611a 0.000513c 0.611a 0.000445 0.611a 0.000377 0.612a

INTEL 0.000896b 1.096 0.000506 1.420a 0.000455 1.420a 0.000462 1.181b 0.000441 1.181b

INTL. BUS. MACH. 0.000294 0.772b 0.000214 0.772b 0.000115 0.704a 0.000109 0.704a 0.000104 0.704a

INTERNATIONAL PAPER 0.000675a 0.932 0.000106 0.927 -2.36E-05 0.83 -0.000118 0.973 -0.000164 1.134

J.P. MORGAN CHASE & CO. 0.001124a 0.792b 0.000496c 0.791b 0.000389 1.027 0.000308 1.028 0.000243 0.819c

JOHNSON & JOHNSON 0.000795a 0.769a 0.000328 0.746b 0.000277 0.746b 0.000193 0.746b 0.000148 1.183a

MCDONALDS 0.000597b 0.893 0.000292 0.711a 0.000273 0.711a 0.000241 0.711a 0.00021 0.730a

MERCK 0.000818a 0.917 0.000412c 0.916 0.000329 0.887b 0.000316 0.888b 0.000303 0.888b

MICROSOFT 0.001735a 0.891 0.001342b 1.025 0.001255b 1.026 0.001036b 1.518b 0.001029b 1.518b

MINNESOTA MNG. & MNFG. 0.000536a 0.896 0.000398b 0.896 0.000191 0.897 -4.32E-06 0.882 -8.65E-05 0.977

PHILIP MORRIS 0.000522c 0.854c 0.000421c 0.709a 0.000372 1.154b 0.000366 1.153b 0.000361 1.153b

PROCTER & GAMBLE 0.000494b 0.821b 0.0002 1.221 0.000149 1.221 0.000115 0.851b 9.42E-05 0.852b

SBC COMMUNICATIONS 0.000602b 0.577a 0.000543b 0.577a 0.000455c 0.577a 0.000308 0.578a 0.00011 1.121

SEARS, ROEBUCK & CO. 0.000513b 0.88 0.000155 0.714a 0.000153 0.715a 0.00015 0.715a 0.000146 0.716a

UNITED TECHNOLOGIES 0.001255a 0.868 0.000600b 0.868 0.000437b 0.648a 0.000375b 0.648a 0.000303c 1.021

WAL - MART STORES 0.001106a 1.119 0.000854a 1.118 0.000714b 1.209 0.000683b 1.21 0.000663b 1.192c

WALT DISNEY 0.000983a 0.93 0.000363 0.742b 0.000302 1.274a 0.000302 1.274a 0.000301 1.274a

Table 3.10: Testing for predictive ability: mean return criterion. Nominal (p_n), White's (2000) Reality Check (p_W) and Hansen's (2001) Superior Predictive Ability (p_H) p-values, when strategies are ranked by the mean return criterion, for the period 1973:1-2001:6 in the case of 0 and 0.10% costs per trade, for the period 1973:1-1986:12 in the case of 0 and 0.25% costs per trade and for the period 1987:1-2001:6 only in the case of 0% costs per trade.

period 1973:1-2001:6 1973:1-1986:12 1987:1-2001:6

costs per trade 0% 0.10% 0% 0.25% 0%

Data set p_n p_W p_H p_n p_W p_H p_n p_W p_H p_n p_W p_H p_n p_W p_H

DJIA 0 0.92 0.06 0 1 0.85 0 0.91 0.03 0 1 0.52 0 0.92 0.1

ALCOA 0 0.86 0 0 0.99 0.45 0 0.82 0.01 0.01 1 0.95 0 0.82 0.2

Table 3.10 continued.

period 1973:1-2001:6 1973:1-1986:12 1987:1-2001:6

costs per trade 0% 0.10% 0% 0.25% 0%

Data set p_n p_W p_H p_n p_W p_H p_n p_W p_H p_n p_W p_H p_n p_W p_H

AMERICAN EXPRESS 0 0.81 0.54 0 1 0.97 0 0.01 0.01 0 1 0.82 0 1 0.99

AT&T 0 1 0.98 0 1 0.98 0.01 1 0.84 0 1 0.96 0 1 0.96

BETHLEHEM STEEL 0 1 0.73 0 1 0.98 0 1 0.22 0 1 0.98 0 1 1

BOEING 0 0.97 0.01 0.03 1 0.97 0 0.98 0.09 0 1 0.96 0 0.96 0.47

CATERPILLAR 0 0.85 0.44 0 0.98 0.92 0 0.85 0.11 0 1 0.98 0 0.99 0.92

CHEVRON - TEXACO 0 0.96 0.09 0 1 0.65 0 0.92 0 0 1 0.75 0 1 0.77

CITIGROUP 0 1 0.69 0 1 0.87 0 1 0.69

COCA - COLA 0 0.98 0.83 0 1 0.99 0 0.99 0.48 0 1 0.98 0 1 0.99

E.I. DU PONT DE NEMOURS 0 0.99 0.47 0 1 0.87 0 0.92 0.28 0 1 0.99 0 0.78 0.14

EASTMAN KODAK 0 1 0.98 0 1 0.99 0 1 0.44 0 1 0.46 0 1 1

EXXON MOBIL 0 1 0.92 0 1 0.98 0 0.96 0.46 0 1 0.95 0 1 0.69

GENERAL ELECTRIC 0.02 1 0.94 0 1 0.99 0 0.94 0.76 0 1 0.95 0 1 0.96

GENERAL MOTORS 0.01 1 0.98 0 1 0.98 0 0.9 0.82 0 1 0.94 0 1 0.78

GOODYEAR TIRE 0 1 0.8 0 1 0.99 0 0.99 0.35 0 1 0.59 0 1 0.98

HEWLETT - PACKARD 0 1 0.98 0 1 0.96 0 1 0.83 0 1 0.99 0 1 0.98

HOME DEPOT 0 1 0.38 0 1 0.85 0 1 0.27

HONEYWELL INTL. 0 1 0.97 0 1 0.97 0 1 0.97

INTEL 0.02 1 1 0 1 1 0 1 0.99 0 1 1 0.08 1 1

INTL. BUS. MACH. 0 1 0.81 0 1 0.91 0 1 0.96 0 1 0.96 0 1 0.79

INTERNATIONAL PAPER 0 0.75 0.16 0.02 1 1 0 0.66 0 0.02 1 0.99 0 1 0.94

J.P. MORGAN CHASE & CO. 0 1 0 0 1 0.84 0 0.88 0.06 0 1 0.94 0 1 0.77

JOHNSON & JOHNSON 0 0.85 0.06 0 1 0.98 0 0.7 0.27 0 1 0.98 0 1 0.94

MCDONALDS 0.02 1 0.44 0 1 0.99 0 1 0.04 0 1 0.96 0 1 0.96

MERCK 0 0.96 0.04 0 1 0.86 0 0.9 0.01 0 1 1 0 1 0.58

MICROSOFT 0 1 0.97 0 1 1 0 1 0.97

MINNESOTA MNG. & MNFG. 0 0.97 0.12 0 1 0.42 0 1 0.08 0 1 0.93 0 0.76 0.04

PHILIP MORRIS 0.08 1 0.88 0 1 0.96 0 0.81 0.25 0 1 0.71 0 1 0.99

PROCTER & GAMBLE 0 1 0.84 0 1 1 0 1 0.93 0 1 1 0.02 1 1

SBC COMMUNICATIONS 0 0.98 0.54 0 1 0.66 0 1 0.76

SEARS, ROEBUCK & CO. 0 1 0.29 0 1 0.98 0 1 0.01 0 1 0.8 0 1 0.99

UNITED TECHNOLOGIES 0 0.95 0 0 1 0.76 0 0.99 0 0 1 0.8 0 0.38 0.28

WAL - MART STORES 0 0.98 0.34 0 1 0.89 0 0.72 0 0 1 0.22 0 1 0.75

WALT DISNEY 0 1 0.31 0 1 1 0 1 0.76 0 1 1 0.07 1 0.97

Table 3.12: Statistics best strategy: 1973-2001, Sharpe ratio criterion, 0 and 0.25% costs. Statistics of the best strategy, selected by the Sharpe ratio criterion, if 0 and 0.25% costs per trade are implemented, for each data series listed in the first column. Column 2 shows the strategy parameters. Columns 3 and 4 show the mean return and excess mean return on a yearly basis in %/100 terms. Columns 5 and 6 show the Sharpe and excess Sharpe ratio. Column 7 shows the largest cumulative loss of the strategy in %/100 terms. Columns 8, 9 and 10 show the number of trades, the percentage of profitable trades and the percentage of days these profitable trades lasted. The last column shows the standard deviation of returns during profitable trades divided by the standard deviation of returns during non-profitable trades. Results are only shown for those data series for which a different best strategy is selected by the Sharpe ratio criterion than by the mean return criterion.

0% costs per trade

Data set Strategy parameters r r^e S S^e ML # tr. %tr.>0 %d > 0 SDR

CITIGROUP [ FR: 0.005, 0, 5 ] 0.5350 0.2047 0.0542 0.0208 -0.6571 786 0.659 0.749 1.1370

HOME DEPOT [ MA: 1, 2, 0.000, 0, 0, 0.000] 0.5925 0.2019 0.0497 0.0128 -0.7714 1993 0.685 0.787 1.2589

MERCK [ MA: 1, 2, 0.001, 0, 0, 0.000] 0.3381 0.1895 0.0390 0.0283 -0.5995 2877 0.696 0.807 1.2504

PHILIP MORRIS [ FR: 0.080, 0, 50 ] 0.2094 0.0603 0.0234 0.0107 -0.6697 163 0.755 0.835 0.9455

0.25% costs per trade

Data set Strategy parameters r r^e S S^e ML # tr. %tr.>0 %d > 0 SDR

ALCOA [ FR: 0.090, 0, 25 ] 0.1325 0.0350 0.0100 0.0069 -0.4430 258 0.647 0.688 1.1084

BOEING [ FR: 0.100, 0, 25, ] 0.2346 0.0391 0.0242 0.0046 -0.6514 212 0.679 0.718 0.9502

E.I. DU PONT DE NEMOURS [ MA: 25, 50, 0.000, 0, 50, 0.000] 0.0984 0.0348 0.0039 0.0076 -0.5338 165 0.685 0.745 1.0104

EASTMAN KODAK [ MA: 25, 50, 0.050, 0, 0, 0.000] 0.0416 0.0368 -0.0041 0.0114 -0.9162 16 0.688 0.775 1.0509

GENERAL ELECTRIC [ MA: 2, 200, 0.000, 0, 25, 0.000] 0.1607 0.0224 0.0177 0.0046 -0.5022 108 0.685 0.716 0.9953

HOME DEPOT [ MA: 2, 5, 0.050, 0, 0, 0.000] 0.4571 0.0998 0.0398 0.0029 -0.6544 25 0.680 0.941 0.7174

INTEL [ SR: 150, 0.000, 0, 50, 0.000 ] 0.2737 0.0442 0.0229 0.0055 -0.6109 47 0.723 0.696 0.9037

INTL. BUS. MACH. [ SR: 100, 0.000, 0, 0, 0.050 ] 0.1138 0.0387 0.0074 0.0086 -0.6570 79 0.519 0.601 0.9881

J.P. MORGAN CHASE & CO. [ MA: 1, 100, 0.000, 0, 50, 0.000] 0.1887 0.1089 0.0166 0.0176 -0.7393 149 0.698 0.725 0.9109

MICROSOFT [ MA: 2, 100, 0.000, 0, 10, 0.000] 0.4528 0.0495 0.0485 0.0060 -0.6099 133 0.481 0.720 0.7916

PHILIP MORRIS [ FR: 0.080, 0, 50 ] 0.1924 0.0455 0.0206 0.0078 -0.6714 163 0.748 0.833 0.9402

PROCTER & GAMBLE [ FR: 0.120, 0, 50 ] 0.1238 0.0328 0.0105 0.0081 -0.4245 94 0.734 0.719 1.2723

WAL - MART STORES [ SR: 200, 0.025, 0, 0, 0.000 ] 0.3531 0.0566 0.0366 0.0012 -0.5933 5 1.000 1.000 NA

WALT DISNEY [ SR: 250, 0.050, 0, 0, 0.000 ] 0.1595 0.0265 0.0182 0.0091 -0.5259 4 0.750 0.999 0.5323

Table 3.13: Sharpe ratio best strategy in excess of Sharpe ratio buy-and-hold. Sharpe ratio of the best strategy, selected by the Sharpe ratio criterion after implementing 0, 0.10 , 0.25 and 0.75% costs per trade, in excess of the Sharpe ratio of the buy-and-hold benchmark, for the full sample period 1973:1-2001:6 and the two subperiods 1973:1-1986:12 and 1987:1-2001:6, for each data series listed in the first column.

period 1973:1-2001:6 1973:1-1986:12 1987:1-2001:6

Data set 0% 0.10% 0.25% 0.75% 0% 0.10% 0.25% 0.75% 0% 0.10% 0.25% 0.75%

DJIA 0.0385 0.0156 0.0153 0.0141 0.0557 0.0317 0.0314 0.0304 0.0279 0.0135 0.0104 0.0096

ALCOA 0.0474 0.0240 0.0069 0.0055 0.0698 0.0513 0.0271 0.0115 0.0302 0.0255 0.0186 0.0032

AMERICAN EXPRESS 0.0270 0.0132 0.0113 0.0059 0.0756 0.0515 0.0277 0.0162 0.0179 0.0159 0.0143 0.0133

AT&T 0.0148 0.0131 0.0123 0.0118 0.0327 0.0201 0.0201 0.0201 0.0179 0.0164 0.0153 0.0140

BETHLEHEM STEEL 0.0389 0.0282 0.0255 0.0187 0.0728 0.0487 0.0372 0.0307 0.0313 0.0298 0.0274 0.0195

BOEING 0.0277 0.0066 0.0046 0.0006 0.0387 0.0188 0.0093 0.0080 0.0259 0.0164 0.0135 0.0037

CATERPILLAR 0.0569 0.0326 0.0093 0.0008 0.0919 0.0646 0.0342 0.0153 0.0320 0.0255 0.0219 0.0100

CHEVRON - TEXACO 0.0290 0.0172 0.0129 0.0046 0.0650 0.0351 0.0250 0.0123 0.0209 0.0192 0.0166 0.0079

CITIGROUP 0.0208 0.0162 0.0136 0.0125 0.0208 0.0162 0.0136 0.0125

COCA - COLA 0.0254 0.0151 0.0150 0.0146 0.0501 0.0289 0.0289 0.0286 0.0120 0.0120 0.0119 0.0117

E.I. DU PONT DE NEMOURS 0.0238 0.0170 0.0076 0.0056 0.0500 0.0309 0.0208 0.0193 0.0341 0.0231 0.0132 0.0062

EASTMAN KODAK 0.0154 0.0127 0.0114 0.0108 0.0347 0.0339 0.0327 0.0286 0.0242 0.0186 0.0138 0.0044

EXXON MOBIL 0.0106 0.0074 0.0072 0.0063 0.0393 0.0205 0.0202 0.0193 0.0175 0.0157 0.0131 0.0042

GENERAL ELECTRIC 0.0132 0.0061 0.0046 0.0021 0.0377 0.0236 0.0208 0.0117 0.0105 0.0102 0.0096 0.0077

GENERAL MOTORS 0.0198 0.0181 0.0179 0.0173 0.0514 0.0388 0.0385 0.0376 0.0277 0.0270 0.0258 0.0218

GOODYEAR TIRE 0.0280 0.0166 0.0166 0.0164 0.0377 0.0349 0.0307 0.0167 0.0401 0.0400 0.0398 0.0392

HEWLETT - PACKARD 0.0136 0.0128 0.0118 0.0082 0.0217 0.0139 0.0122 0.0069 0.0183 0.0177 0.0168 0.0136

HOME DEPOT 0.0128 0.0053 0.0029 0.0011 0.0212 0.0066 0.0054 0.0023

HONEYWELL INTL. 0.0289 0.0277 0.0258 0.0196 0.0265 0.0252 0.0234 0.0171

INTEL 0.0103 0.0059 0.0055 0.0041 0.0366 0.0289 0.0258 0.0215 0.0122 0.0107 0.0086 0.0072

INTL. BUS. MACH. 0.0166 0.0129 0.0086 0.0080 0.0235 0.0212 0.0194 0.0167 0.0217 0.0203 0.0201 0.0196

INTERNATIONAL PAPER 0.0378 0.0149 0.0095 0.0041 0.0831 0.0557 0.0208 0.0090 0.0247 0.0201 0.0139 0.0085

J.P. MORGAN CHASE & CO. 0.0444 0.0210 0.0176 0.0133 0.0622 0.0370 0.0251 0.0185 0.0425 0.0217 0.0206 0.0167

JOHNSON & JOHNSON 0.0315 0.0093 0.0069 0.0023 0.0578 0.0285 0.0261 0.0248 0.0094 0.0062 0.0051 0.0029

MCDONALDS 0.0193 0.0119 0.0109 0.0073 0.0458 0.0294 0.0203 0.0147 0.0137 0.0131 0.0124 0.0098

MERCK 0.0283 0.0094 0.0076 0.0064 0.0591 0.0292 0.0137 0.0134 0.0147 0.0102 0.0097 0.0083

MICROSOFT 0.0165 0.0082 0.0060 0.0028 0.0165 0.0082 0.0060 0.0028

MINNESOTA MNG. & MNFG. 0.0330 0.0259 0.0153 0.0011 0.0637 0.0335 0.0307 0.0215 0.0362 0.0293 0.0190 0.0026

PHILIP MORRIS 0.0107 0.0095 0.0078 0.0065 0.0324 0.0220 0.0200 0.0131 0.0177 0.0152 0.0115 0.0083

PROCTER & GAMBLE 0.0236 0.0098 0.0081 0.0054 0.0439 0.0245 0.0191 0.0183 0.0157 0.0146 0.0129 0.0093

SBC COMMUNICATIONS 0.0243 0.0213 0.0169 0.0029 0.0232 0.0203 0.0160 0.0023

SEARS, ROEBUCK & CO. 0.0305 0.0167 0.0166 0.0164 0.0653 0.0331 0.0264 0.0257 0.0202 0.0185 0.0173 0.0137

UNITED TECHNOLOGIES 0.0414 0.0135 0.0115 0.0046 0.0403 0.0187 0.0161 0.0100 0.0424 0.0182 0.0149 0.0050

Table 3.13 continued.

period 1973:1-2001:6 1973:1-1986:12 1987:1-2001:6

Data set 0% 0.10% 0.25% 0.75% 0% 0.10% 0.25% 0.75% 0% 0.10% 0.25% 0.75%

WAL - MART STORES 0.0144 0.0052 0.0012 0.0008 0.0476 0.0380 0.0235 0.0064 0.0164 0.0076 0.0052 -0.0009

WALT DISNEY 0.0288 0.0092 0.0091 0.0089 0.0410 0.0183 0.0163 0.0158 0.0182 0.0174 0.0162 0.0123

Average 0.0258 0.0146 0.0112 0.0078 0.0509 0.0322 0.0240 0.0181 0.0229 0.0179 0.0153 0.0100

Table 3.14: Estimation results CAPM: 1973-2001, Sharpe ratio criterion. Coefficient estimates of the Sharpe-Lintner CAPM in the period January 1, 1973 through June 29, 2001: r_tⁱ-r_t^f=a + b (r_t^DJIA-r_t^f) + e_t. That is, the excess return of the best strategy, selected by the Sharpe ratio criterion, over the risk-free interest rate is regressed against a constant and the excess return of the DJIA over the risk-free interest rate. Estimation results for the 0, 0.10, 0.25, 0.50 and 0.75% costs per trade cases are shown. a, b, c indicates that the corresponding coefficient is, in the case of a, significantly different from zero, or in the case of b, significantly different from one, at the 1, 5, 10% significance level. Estimation is done with Newey-West (1987) heteroskedasticity and autocorrelation consistent (HAC) standard errors.

costs per trade 0% 0.10% 0.25% 0.50% 0.75%

Data set a b a b a b a b a b

DJIA 0.000539a 0.872c 0.000210b 0.837 0.000206b 0.837c 0.000199b 0.838c 0.000191c 0.839c

ALCOA 0.001299a 0.887 0.000677b 0.885 0.000158 0.738a 0.000188 1.207a 0.000177 1.207a

AMERICAN EXPRESS 0.000762a 1.150c 0.000352 1.06 0.000301 1.06 0.000215 1.061 0.000144 1.228a

AT&T 0.000125 0.681a 9.18E-05 0.681a 6.59E-05 0.87 6.05E-05 0.87 5.51E-05 0.87

BETHLEHEM STEEL 0.000417 0.829c 4.16E-05 0.434a -3.90E-05 0.868 -0.00017 0.867 -0.000301 0.867

BOEING 0.001248a 0.879 0.000540b 0.83 0.000496b 0.83 0.000422c 0.83 0.000444c 1.215a

CATERPILLAR 0.001245a 0.781b 0.000648b 0.773a 5.32E-05 1.264 -9.66E-05 1.263 -0.000178 1.195

CHEVRON - TEXACO 0.000630a 0.667a 0.000376 0.744a 0.000282 0.744a 0.000187 0.672a 0.000101 0.672a

CITIGROUP 0.001185a 1.114 0.001045a 0.926 0.000962a 1.701a 0.000947a 1.701a 0.000931a 1.701a

COCA - COLA 0.000651a 0.832b 0.000308c 0.782a 0.000307c 0.782a 0.000304c 0.783a 0.000301c 0.783a

E.I. DU PONT DE NEMOURS 0.000406c 0.872 0.000238 0.918 3.17E-05 0.816c -1.70E-05 1.108 -1.79E-05 1.108

EASTMAN KODAK -4.80E-05 0.918 -0.000114 0.918 -0.000212 1.431c -0.000222 1.431c -0.000232 1.431c

EXXON MOBIL 0.00029 1.027 0.000217 0.956 0.000211 0.956 0.000202 0.956 0.000193 0.956

GENERAL ELECTRIC 0.000499b 1.017 0.000307c 0.790c 0.000242c 0.988 0.000302 1.214 0.00028 1.215

GENERAL MOTORS 0.000264 0.861 0.000206 0.674a 0.000202 0.674a 0.000195 0.674a 0.000188 0.674a

GOODYEAR TIRE 0.000519c 0.789a 0.000234 0.603a 0.000232 0.604a 0.000229 0.605a 0.000226 0.606a

HEWLETT - PACKARD 0.000474c 1.083 0.000455c 1.083 0.000428c 1.083 0.000382 1.082 0.000336 1.082

HOME DEPOT 0.001339a 0.977 0.001082b 1.578c 0.000955b 1.102 0.000927b 1.102 0.000900b 1.103

HONEYWELL INTL. 0.000581c 0.611a 0.000554c 0.611a 0.000513c 0.611a 0.000445 0.611a 0.000377 0.612a

INTEL 0.000896b 1.096 0.000494 1.181b 0.000482 1.181b 0.000462 1.181b 0.000441 1.181b

INTL. BUS. MACH. 0.000294 0.772b 0.000214 0.772b 8.23E-05 0.916 0.000109 0.704a 0.000104 0.704a

Table 3.14 continued.

costs per trade 0% 0.10% 0.25% 0.50% 0.75%

Data set a b a b a b a b a b

INTERNATIONAL PAPER 0.000675a 0.932 0.000106 0.927 -2.36E-05 0.83 -0.000118 0.973 -0.000164 1.134

J.P. MORGAN CHASE & CO. 0.001124a 0.792b 0.000496c 0.791b 0.000345 0.819c 0.000294 0.819c 0.000243 0.819c

JOHNSON & JOHNSON 0.000795a 0.769a 0.000328 0.746b 0.000277 0.746b 0.000193 0.746b 0.000148 1.183a

MCDONALDS 0.000597b 0.893 0.000292 0.711a 0.000273 0.711a 0.000241 0.711a 0.000209 0.711a

MERCK 0.000814a 0.848 0.000412c 0.916 0.000329 0.887b 0.000316 0.888b 0.000303 0.888b

MICROSOFT 0.001735a 0.891 0.001111a 1.063 0.001057a 1.062 0.000924a 1.200a 0.000917a 1.201a

MINNESOTA MNG. & MNFG. 0.000536a 0.896 0.000398b 0.896 0.000191 0.897 -4.32E-06 0.882 -8.67E-05 0.882

PHILIP MORRIS 0.000420b 0.702a 0.000398b 0.702a 0.000364c 0.702a 0.000310c 0.915b 0.000308c 0.915b

PROCTER & GAMBLE 0.000494b 0.821b 0.000156 0.617a 0.000133 0.636a 0.0001 0.636a 7.45E-05 0.954

SBC COMMUNICATIONS 0.000602b 0.577a 0.000543b 0.577a 0.000455c 0.577a 0.000308 0.578a 0.00011 1.121

SEARS, ROEBUCK & CO. 0.000513b 0.88 0.000155 0.714a 0.000153 0.715a 0.00015 0.715a 0.000146 0.716a

UNITED TECHNOLOGIES 0.001255a 0.868 0.000474b 0.648a 0.000437b 0.648a 0.000375b 0.648a 0.000314c 0.648a

WAL - MART STORES 0.001106a 1.119 0.000854a 1.118 0.000670b 1.192c 0.000666b 1.192c 0.000663b 1.192c

WALT DISNEY 0.000983a 0.93 0.000253 0.731c 0.000252 0.731c 0.00025 0.731c 0.000248 0.731c

Table 3.16: Testing for predictive ability: Sharpe ratio criterion. Nominal (p_n), White's (2000) Reality Check (p_W) and Hansen's (2001) Superior Predictive Ability (p_H) p-values, when strategies are ranked by the Sharpe ratio criterion, for the full sample period 1973:1-2001:6 in the case of 0 and 0.10% costs per trade and for the subperiods 1973:1-1986:12 and 1987:1-2001:6 in the case of 0 and 0.25% costs per trade.

period 1973:1-2001:6 1973:1-1986:12 1987:1-2001:6

costs per trade 0% 0.10% 0% 0.25% 0% 0.25%

Data set p_n p_W p_H p_n p_W p_H p_n p_W p_H p_n p_W p_H p_n p_W p_H p_n p_W p_H

DJIA 0 0.21 0.06 0.05 1 0.46 0 0.24 0.05 0.01 0.97 0.26 0 0.93 0.28 0.04 1 0.7

ALCOA 0 0.02 0 0 0.86 0.04 0 0.02 0 0 0.98 0.29 0 0.93 0.12 0 1 0.37

AMERICAN EXPRESS 0 0.54 0.06 0 0.99 0.44 0 0.02 0 0 0.99 0.29 0 1 0.56 0 1 0.52

AT&T 0 1 0.47 0 1 0.46 0.02 0.71 0.21 0.05 1 0.41 0 1 0.69 0 1 0.64

BETHLEHEM STEEL 0 0.18 0.05 0 0.55 0.14 0 0.03 0.02 0 0.72 0.35 0 0.82 0.36 0 0.91 0.35

BOEING 0 0.74 0.01 0 1 0.77 0 0.94 0.01 0 1 0.8 0.01 0.99 0.23 0 1 0.68

CATERPILLAR 0 0.04 0.01 0 0.42 0.04 0 0.02 0.01 0 0.88 0.27 0 0.91 0.16 0 1 0.3

CHEVRON - TEXACO 0.01 0.49 0.04 0 0.98 0.17 0 0.02 0 0 0.99 0.25 0 1 0.26 0 1 0.28

CITIGROUP 0 1 0.24 0 1 0.3 0 1 0.24 0 1 0.33

COCA - COLA 0 0.66 0.06 0.02 1 0.22 0 0.36 0.06 0.02 0.97 0.3 0 1 0.81 0 1 0.51

E.I. DU PONT DE NEMOURS 0.01 0.79 0.15 0 0.97 0.27 0 0.24 0.06 0.01 0.99 0.58 0 0.69 0.03 0 1 0.56

Table 3.16 continued.

period 1973:1-2001:6 1973:1-1986:12 1987:1-2001:6

costs per trade 0% 0.10% 0% 0.25% 0% 0.25%

Data set p_n p_W p_H p_n p_W p_H p_n p_W p_H p_n p_W p_H p_n p_W p_H p_n p_W p_H

EASTMAN KODAK 0 0.98 0.56 0 1 0.66 0 0.64 0.18 0 0.74 0.16 0 0.99 0.39 0 1 0.73

EXXON MOBIL 0.03 1 0.44 0.04 1 0.5 0 0.6 0.08 0.01 1 0.35 0 1 0.17 0 1 0.21

GENERAL ELECTRIC 0.05 1 0.36 0.01 1 0.69 0.01 0.65 0.11 0 1 0.37 0 1 0.76 0 1 0.56

GENERAL MOTORS 0.02 0.96 0.26 0.04 0.98 0.26 0 0.35 0.08 0.02 0.74 0.17 0 0.94 0.19 0 0.96 0.16

GOODYEAR TIRE 0 0.72 0.13 0.05 1 0.52 0 0.75 0.27 0 0.96 0.28 0 0.68 0.05 0 0.68 0.03

HEWLETT - PACKARD 0 1 0.37 0 1 0.3 0 1 0.57 0.01 1 0.77 0 1 0.34 0 1 0.24

HOME DEPOT 0.09 1 0.71 0.07 1 0.92 0.01 1 0.22 0.01 1 0.76

HONEYWELL INTL. 0 0.96 0.12 0 0.98 0.12 0 0.98 0.2 0 1 0.19

INTEL 0.08 1 0.65 0.01 1 0.81 0 0.95 0.22 0 1 0.42 0 1 0.76 0 1 0.72

INTL. BUS. MACH. 0 0.96 0.44 0 0.99 0.57 0 0.98 0.58 0.01 1 0.55 0 0.99 0.58 0 1 0.41

INTERNATIONAL PAPER 0 0.16 0.03 0.02 1 0.49 0 0.02 0.01 0.01 1 0.5 0 0.98 0.36 0 1 0.62

J.P. MORGAN CHASE & CO. 0 0.04 0.01 0 0.88 0.18 0 0.12 0.02 0 0.98 0.42 0 0.42 0.01 0 1 0.4

JOHNSON & JOHNSON 0 0.28 0.03 0 1 0.53 0 0.09 0.04 0 0.92 0.33 0 1 0.86 0.01 1 0.84

MCDONALDS 0.03 0.96 0.27 0 1 0.48 0 0.35 0.08 0 0.99 0.56 0 1 0.76 0 1 0.54

MERCK 0 0.66 0.06 0 1 0.57 0 0.08 0.02 0.07 1 0.77 0 1 0.64 0 1 0.63

MICROSOFT 0.01 1 0.26 0 1 0.74 0.01 1 0.26 0 1 0.77

MINNESOTA MNG. & MNFG. 0 0.3 0.04 0 0.7 0.06 0 0.04 0.02 0 0.88 0.18 0 0.56 0 0 1 0.23

PHILIP MORRIS 0 1 0.66 0 1 0.47 0.01 0.81 0.22 0 1 0.33 0 1 0.53 0 1 0.58

PROCTER & GAMBLE 0 0.74 0.12 0 1 0.66 0.01 0.54 0.1 0 1 0.62 0 1 0.74 0 1 0.65

SBC COMMUNICATIONS 0 0.99 0.15 0 1 0.14 0 1 0.21 0 1 0.27

SEARS, ROEBUCK & CO. 0 0.36 0.05 0.04 0.98 0.39 0 0.03 0.02 0.02 0.99 0.4 0 1 0.5 0 1 0.45

UNITED TECHNOLOGIES 0 0.1 0.01 0 1 0.35 0 0.76 0.09 0 1 0.64 0 0.57 0.03 0 1 0.58

WAL - MART STORES 0 1 0.24 0.02 1 0.72 0 0.95 0.02 0 1 0.29 0 1 0.39 0 1 0.83

WALT DISNEY 0 0.55 0.04 0.04 1 0.81 0.01 0.66 0.18 0.04 1 0.9 0 1 0.59 0 1 0.39

Table 3.18: Statistics best out-of-sample testing procedure: mean return criterion, 0.25% costs. Statistics of the best recursively optimizing and testing procedure applied to each data series listed in the first column in the case of 0.25% costs per trade. The best strategy in the optimizing period is selected on the basis of the mean return criterion. Column 2 shows the sample period. Column 3 shows the parameters: [length optimizing period, length testing period]. Columns 4 and 5 show the mean return and excess mean return on a yearly basis in %/100 terms. Columns 6 and 7 show the Sharpe and excess Sharpe ratio. Column 8 shows the largest cumulative loss in %/100 terms. Columns 9, 10 and 11 show the number of trades, the percentage of profitable trades and the percentage of days these profitable trades lasted. The last column shows the standard deviation of returns during profitable trades divided by the standard deviation of returns during non-profitable trades.

Data set Period Parameters r r^e S S^e ML # tr %tr>0 %d>0 SDR

DJIA 10/31/78-6/29/01 [ 1008, 252 ] 0.1349 0.0198 0.0146 0.0018 -0.4311 164 0.494 0.881 1.0635

ALCOA 10/31/78-6/29/01 [ 1260, 126 ] 0.2044 0.0881 0.0172 0.0102 -0.6114 1278 0.292 0.541 1.1771

AMERICAN EXPRESS 10/31/78-6/29/01 [ 756, 126 ] 0.1247 0.0011 0.0059 -0.0018 -0.7433 378 0.526 0.693 1.0498

AT&T 10/31/78-6/29/01 [ 1260, 252 ] 0.0145 -0.0359 -0.0097 -0.0035 -0.9082 182 0.626 0.738 0.9071

BETHLEHEM STEEL 10/31/78-6/29/01 [ 1008, 252 ] 0.1782 0.2726 0.0107 0.0350 -0.7102 917 0.266 0.610 1.1660

BOEING 10/31/78-6/29/01 [ 126, 10 ] 0.1240 0.0054 0.0063 -0.0012 -0.6747 678 0.409 0.629 0.9991

CATERPILLAR 10/31/78-6/29/01 [ 1260, 252 ] 0.2062 0.1502 0.0168 0.0213 -0.5878 1473 0.249 0.501 1.0641

CHEVRON - TEXACO 10/31/78-6/29/01 [ 1008, 126 ] 0.1195 0.0251 0.0070 0.0036 -0.5047 736 0.327 0.641 1.1284

CITIGROUP 8/26/92-6/29/01 [ 42, 10 ] 0.4220 0.0727 0.0369 -0.0068 -0.5172 321 0.371 0.541 1.1087

COCA - COLA 10/31/78-6/29/01 [ 252, 126 ] 0.1577 0.0080 0.0123 -0.0028 -0.7893 555 0.404 0.658 1.0912

E.I. DU PONT DE NEMOURS 10/31/78-6/29/01 [ 1260, 126 ] 0.0293 -0.0563 -0.0081 -0.0096 -0.7602 598 0.426 0.568 0.9584

EASTMAN KODAK 10/31/78-6/29/01 [ 1260, 252 ] 0.0055 -0.0297 -0.0130 -0.0039 -0.7596 288 0.462 0.731 0.9212

EXXON MOBIL 10/31/78-6/29/01 [ 1260, 126 ] 0.1135 -0.0060 0.0057 -0.0046 -0.6069 219 0.516 0.733 1.1681

GENERAL ELECTRIC 10/31/78-6/29/01 [ 756, 252 ] 0.1743 -0.0052 0.0142 -0.0087 -0.6596 187 0.588 0.811 0.8271

GENERAL MOTORS 10/31/78-6/29/01 [ 504, 252 ] 0.0700 0.0252 -0.0013 0.0058 -0.6637 353 0.439 0.676 0.8982

GOODYEAR TIRE 10/31/78-6/29/01 [ 252, 126 ] 0.1014 0.0463 0.0031 0.0076 -0.8122 503 0.408 0.636 1.0959

HEWLETT - PACKARD 10/31/78-6/29/01 [ 504, 252 ] 0.1656 0.0101 0.0096 -0.0019 -0.8086 467 0.493 0.637 1.0680

HOME DEPOT 10/30/89-6/29/01 [ 126, 42 ] 0.4145 0.0964 0.0403 0.0002 -0.4604 346 0.376 0.620 0.8692

HONEYWELL INTL. 7/18/91-6/29/01 [ 1260, 252 ] 0.1958 0.0609 0.0166 0.0020 -0.6052 87 0.494 0.806 0.6487

INTEL 10/29/84-6/29/01 [ 252, 126 ] 0.3172 0.0611 0.0236 0.0001 -0.6988 419 0.372 0.672 1.1498

INTL. BUS. MACH. 10/31/78-6/29/01 [ 1260, 126 ] 0.0348 -0.0490 -0.0061 -0.0071 -0.8511 154 0.571 0.753 0.9488

INTERNATIONAL PAPER 10/31/78-6/29/01 [ 1260, 126 ] 0.0871 0.0327 0.0011 0.0060 -0.7551 884 0.309 0.561 0.9877

J.P. MORGAN CHASE & CO. 10/31/78-6/29/01 [ 756, 252 ] 0.2001 0.1104 0.0145 0.0126 -0.7584 441 0.358 0.673 1.0335

JOHNSON & JOHNSON 10/31/78-6/29/01 [ 1008, 252 ] 0.1089 -0.0505 0.0042 -0.0135 -0.6874 291 0.495 0.803 0.9719

MCDONALDS 10/31/78-6/29/01 [ 252, 126 ] 0.1173 -0.0246 0.0063 -0.0074 -0.5120 463 0.445 0.630 1.0525

MERCK 10/31/78-6/29/01 [ 252, 126 ] 0.1658 -0.0036 0.0136 -0.0065 -0.8547 469 0.458 0.748 0.9708

MICROSOFT 1/09/92-6/29/01 [ 21, 10 ] 0.4215 0.1123 0.0385 0.0009 -0.5474 306 0.366 0.538 0.9815

MINNESOTA MNG. & MNFG. 10/31/78-6/29/01 [ 126, 63 ] 0.0758 -0.0172 -0.0006 -0.0041 -0.6431 517 0.435 0.600 1.1240

PHILIP MORRIS 10/31/78-6/29/01 [ 252, 10 ] 0.1660 0.0026 0.0128 -0.0040 -0.6785 450 0.411 0.661 0.9167

PROCTER & GAMBLE 10/31/78-6/29/01 [ 756, 126 ] 0.0629 -0.0491 -0.0027 -0.0101 -0.5250 302 0.520 0.716 0.9495

Table 3.18 continued.

Data set Period Parameters r r^e S S^e ML # tr %tr>0 %d>0 SDR

SBC COMMUNICATIONS 9/18/89-6/29/01 [ 252, 42 ] 0.1027 0.0106 0.0071 -0.0007 -0.6655 260 0.442 0.624 0.8597

SEARS, ROEBUCK & CO. 10/31/78-6/29/01 [ 1008, 252 ] 0.0683 -0.0066 -0.0014 -0.0006 -0.7466 685 0.236 0.617 1.0365

UNITED TECHNOLOGIES 10/31/78-6/29/01 [ 1260, 252 ] 0.1123 -0.0112 0.0054 -0.0042 -0.5593 663 0.330 0.710 0.9482

WAL - MART STORES 10/30/86-6/29/01 [ 252, 21 ] 0.2833 0.0778 0.0275 0.0024 -0.7779 315 0.432 0.655 1.0760

WALT DISNEY 10/31/78-6/29/01 [ 1008, 126 ] 0.1964 0.0301 0.0163 0.0005 -0.6901 291 0.509 0.786 1.0927

Table 3.19: Statistics best out-of-sample testing procedure: Sharpe ratio criterion, 0.25% costs. Statistics of the best recursively optimizing and testing procedure applied to each data series listed in the first column in the case of 0.25% costs per trade. The best strategy in the optimizing period is selected on the basis of the Sharpe ratio criterion. Column 2 shows the sample period. Column 3 shows the parameters: [length optimizing period, length testing period]. Columns 4 and 5 show the mean return and excess mean return on a yearly basis in %/100 terms. Columns 6 and 7 show the Sharpe and excess Sharpe ratio. Column 8 shows the largest cumulative loss in %/100 terms. Columns 9, 10 and 11 show the number of trades, the percentage of profitable trades and the percentage of days these profitable trades lasted. The last column shows the standard deviation of returns during profitable trades divided by the standard deviation of returns during non-profitable trades.

Data set Period Parameters r r^e S S^e ML # tr %tr>0 %d>0 SDR

DJIA 10/31/78-6/29/01 [ 1260, 252 ] 0.1661 0.0510 0.0252 0.0124 -0.3494 85 0.471 0.854 0.6790

ALCOA 10/31/78-6/29/01 [ 1008, 252 ] 0.2202 0.1039 0.0190 0.0120 -0.5396 468 0.177 0.626 1.2539

AMERICAN EXPRESS 10/31/78-6/29/01 [ 756, 126 ] 0.1276 0.0040 0.0063 -0.0014 -0.7372 118 0.644 0.702 1.0477

AT&T 10/31/78-6/29/01 [ 1260, 252 ] -0.0243 -0.0747 -0.0156 -0.0094 -0.9082 62 0.790 0.602 1.0126

BETHLEHEM STEEL 10/31/78-6/29/01 [ 1008, 252 ] 0.1649 0.2593 0.0092 0.0335 -0.6966 422 0.156 0.587 1.1778

BOEING 10/31/78-6/29/01 [ 1260, 252 ] 0.1380 0.0194 0.0079 0.0004 -0.7835 204 0.152 0.664 0.9487

CATERPILLAR 10/31/78-6/29/01 [ 1260, 252 ] 0.2547 0.1987 0.0224 0.0269 -0.5878 593 0.132 0.527 1.0262

CHEVRON - TEXACO 10/31/78-6/29/01 [ 42, 21 ] 0.1111 0.0167 0.0055 0.0021 -0.6538 235 0.251 0.578 1.1345

CITIGROUP 8/26/92-6/29/01 [ 504, 126 ] 0.4496 0.1003 0.0436 -0.0001 -0.6200 66 0.212 0.702 1.0065

COCA - COLA 10/31/78-6/29/01 [ 504, 126 ] 0.1230 -0.0267 0.0077 -0.0074 -0.7367 161 0.248 0.620 1.0784

E.I. DU PONT DE NEMOURS 10/31/78-6/29/01 [ 126, 63 ] 0.1034 0.0178 0.0041 0.0026 -0.5379 149 0.349 0.613 1.0074

EASTMAN KODAK 10/31/78-6/29/01 [ 1008, 252 ] -0.0010 -0.0362 -0.0129 -0.0038 -0.8146 62 0.629 0.705 0.9497

EXXON MOBIL 10/31/78-6/29/01 [ 1260, 126 ] 0.0987 -0.0208 0.0042 -0.0061 -0.4136 57 0.754 0.750 0.8258

GENERAL ELECTRIC 10/31/78-6/29/01 [ 1260, 252 ] 0.1691 -0.0104 0.0159 -0.0070 -0.6553 75 0.853 0.834 0.7889

GENERAL MOTORS 10/31/78-6/29/01 [ 1008, 126 ] 0.0981 0.0533 0.0029 0.0100 -0.6946 90 0.533 0.745 0.9067

GOODYEAR TIRE 10/31/78-6/29/01 [ 63, 1 ] 0.1330 0.0779 0.0082 0.0127 -0.7492 531 0.115 0.477 1.1916

HEWLETT - PACKARD 10/31/78-6/29/01 [ 504, 252 ] 0.1818 0.0263 0.0120 0.0005 -0.8192 152 0.296 0.630 0.9360

HOME DEPOT 10/30/89-6/29/01 [ 252, 42 ] 0.3923 0.0742 0.0426 0.0025 -0.5015 86 0.267 0.653 1.0607

HONEYWELL INTL. 7/18/91-6/29/01 [ 756, 126 ] 0.2393 0.1044 0.0253 0.0107 -0.5554 51 0.314 0.751 0.7963

INTEL 10/29/84-6/29/01 [ 504, 126 ] 0.3872 0.1311 0.0317 0.0082 -0.6771 97 0.423 0.742 0.9992

Table 3.19 continued.

Data set Period Parameters r r^e S S^e ML # tr %tr>0 %d>0 SDR

INTL. BUS. MACH. 10/31/78-6/29/01 [ 252, 126 ] 0.0766 -0.0072 -0.0004 -0.0014 -0.7582 102 0.539 0.665 1.0053

INTERNATIONAL PAPER 10/31/78-6/29/01 [ 756, 252 ] 0.0752 0.0208 -0.0005 0.0044 -0.7404 393 0.176 0.521 1.0723

J.P. MORGAN CHASE & CO. 10/31/78-6/29/01 [ 252, 10 ] 0.1552 0.0655 0.0104 0.0085 -0.8088 278 0.252 0.613 1.0458

JOHNSON & JOHNSON 10/31/78-6/29/01 [ 504, 126 ] 0.1558 -0.0036 0.0134 -0.0043 -0.5298 154 0.351 0.677 0.8972

MCDONALDS 10/31/78-6/29/01 [ 756, 126 ] 0.1543 0.0124 0.0141 0.0004 -0.5308 118 0.576 0.778 0.9448

MERCK 10/31/78-6/29/01 [ 1260, 126 ] 0.1291 -0.0403 0.0078 -0.0123 -0.6230 142 0.373 0.716 0.8361

MICROSOFT 1/09/92-6/29/01 [ 252, 5 ] 0.3537 0.0445 0.0399 0.0023 -0.4535 68 0.279 0.601 1.0901

MINNESOTA MNG. & MNFG. 10/31/78-6/29/01 [ 126, 63 ] 0.0876 -0.0054 0.0018 -0.0017 -0.5439 152 0.441 0.636 1.1148

PHILIP MORRIS 10/31/78-6/29/01 [ 252, 63 ] 0.2714 0.1080 0.0294 0.0126 -0.5213 161 0.373 0.736 1.1326

PROCTER & GAMBLE 10/31/78-6/29/01 [ 1260, 252 ] 0.0642 -0.0478 -0.0028 -0.0102 -0.5686 76 0.421 0.649 0.8839

SBC COMMUNICATIONS 9/18/89-6/29/01 [ 252, 126 ] 0.1133 0.0212 0.0091 0.0013 -0.7219 95 0.274 0.658 0.9916

SEARS, ROEBUCK & CO. 10/31/78-6/29/01 [ 756, 126 ] 0.0868 0.0119 0.0012 0.0020 -0.6139 178 0.191 0.656 1.0114

UNITED TECHNOLOGIES 10/31/78-6/29/01 [ 504, 252 ] 0.1285 0.0050 0.0083 -0.0013 -0.8343 160 0.344 0.723 0.9601

WAL - MART STORES 10/30/86-6/29/01 [ 756, 252 ] 0.2860 0.0805 0.0302 0.0051 -0.4652 72 0.306 0.777 0.8815

WALT DISNEY 10/31/78-6/29/01 [ 1008, 126 ] 0.2076 0.0413 0.0186 0.0028 -0.6100 87 0.552 0.700 1.0510

Table 3.20: Excess performance best out-of-sample testing procedure. Panel A shows the yearly mean return of the best recursive out-of-sample testing procedure, selected by the mean return criterion, in excess of the yearly mean return of the buy-and-hold. Panel B shows the Sharpe ratio of the best recursive out-of-sample testing procedure, selected by the Sharpe ratio criterion, in excess of the Sharpe ratio of the buy-and-hold. Results are presented for the 0, 0.10 and 0.50% transaction costs per trade cases. The results for the 0.25% transaction costs per trade case can be found in the tables 3.18 and 3.19. The row labeled ``Average: out-of-sample'' shows the average over the results as presented in the table. The row labeled ``Average: in sample'' shows the average over the results of the best strategy selected in sample for each data series.

Panel A Panel B

selection criterion Mean return Sharpe ratio

Data set 0% 0.10% 0.50% 0% 0.10% 0.50%

DJIA 0.0775 0.0444 0.0287 0.0189 0.0093 0.0057

ALCOA 0.2881 0.2145 -0.0591 0.0313 0.0187 -0.0045

AMERICAN EXPRESS 0.1020 0.1062 -0.0229 0.0047 0.0052 -0.0024

AT&T 0.0306 -0.0104 -0.0581 0.0038 -0.0027 -0.0048

BETHLEHEM STEEL 0.3270 0.2914 0.1389 0.0472 0.0349 0.0245

BOEING 0.0773 0.0325 -0.0380 0.0125 0.0060 -0.0040

CATERPILLAR 0.3576 0.2667 -0.0567 0.0451 0.0371 0.0043

CHEVRON - TEXACO 0.1262 0.0913 -0.0250 0.0177 0.0111 -0.0038

CITIGROUP 0.1635 0.1384 -0.0176 0.0038 -0.0051 -0.0082

COCA - COLA 0.1576 0.0551 -0.0269 0.0085 0.0041 -0.0117

E.I. DU PONT DE NEMOURS 0.1076 0.0547 -0.0799 0.0174 0.0074 -0.0077

EASTMAN KODAK 0.0529 0.0199 -0.0386 0.0011 0.0021 -0.0041

EXXON MOBIL 0.0200 0.0194 -0.0087 0.0012 -0.0044 -0.0083

GENERAL ELECTRIC 0.0807 0.0295 -0.0407 -0.0028 -0.0071 -0.0028

GENERAL MOTORS 0.0558 0.0286 0.0347 0.0141 0.0109 0.0069

GOODYEAR TIRE 0.1431 0.0982 0.0062 0.0353 0.0261 0.0009

HEWLETT - PACKARD 0.0239 0.0052 0.0078 0.0061 0.0039 -0.0061

HOME DEPOT 0.1915 0.0997 0.0415 0.0009 0.0002 -0.0027

HONEYWELL INTL. 0.2066 0.1337 0.0170 0.0153 0.0118 0.0082

INTEL 0.0660 0.0684 0.0253 0.0137 0.0154 0.0035

INTL. BUS. MACH. -0.0092 -0.0497 -0.0085 0.0042 -0.0011 -0.0012

INTERNATIONAL PAPER 0.1904 0.0871 -0.0439 0.0203 0.0165 -0.0021

J.P. MORGAN CHASE & CO. 0.2582 0.1492 0.0615 0.0406 0.0226 0.0050

JOHNSON & JOHNSON 0.0269 -0.0398 -0.0678 0.0070 -0.0012 -0.0071

MCDONALDS 0.0585 0.0266 -0.0383 0.0087 0.0022 -0.0034

MERCK 0.0731 0.0216 -0.0398 0.0001 -0.0023 -0.0162

MICROSOFT 0.1420 0.1192 0.0658 0.0149 0.0058 -0.0007

MINNESOTA MNG. & MNFG. 0.0891 0.0139 -0.0400 0.0103 0.0031 -0.0073

PHILIP MORRIS 0.0434 0.0162 -0.0246 0.0133 0.0132 -0.0026

PROCTER & GAMBLE 0.0366 0.0147 -0.0543 0.0018 -0.0083 -0.0015

SBC COMMUNICATIONS 0.1295 0.0176 -0.0379 0.0090 0.0015 -0.0015

SEARS, ROEBUCK & CO. 0.0872 0.0354 -0.0165 0.0132 0.0126 0.0032

UNITED TECHNOLOGIES 0.2242 0.0832 -0.0632 0.0354 0.0110 -0.0055

WAL - MART STORES 0.1074 0.0582 0.0284 0.0065 0.0018 -0.0081

WALT DISNEY 0.1841 0.0599 0.0165 0.0247 0.0061 -0.0025

Average: out-of-sample 0.1228 0.0686 -0.0124 0.0145 0.0077 -0.0020

Average: in sample 0.1616 0.1043 0.0676 0.0220 0.0147 0.0091

Table 3.21: Estimation results CAPM for best out-of-sample testing procedure. Coefficient estimates of the Sharpe-Lintner CAPM: r_tⁱ-r_t^f=a + b (r_t^DJIA-r_t^f) + e_t. That is, the return of the best recursive optimizing and testing procedure, when selection is done in the optimizing period by the mean return criterion (Panel A) or by the Sharpe ratio criterion (Panel B), in excess of the risk-free interest rate is regressed against a constant and the return of the DJIA in excess of the risk-free interest rate. Estimation results for the 0 and 0.10% costs per trade cases are shown. a, b, c indicates that the corresponding coefficient is, in the case of a, significantly different from zero, or in the case of b, significantly different from one, at the 1, 5, 10% significance level. Estimation is done with Newey-West (1987) heteroskedasticity and autocorrelation consistent (HAC) standard errors.

Panel A Panel B

selection criterion Mean return Sharpe ratio

costs per trade 0% 0.10% 0% 0.10%

Data set a b a b a b a b

DJIA 0.000311a 0.703a 0.000173 0.901 0.000292a 0.851 0.000165 0.832c

ALCOA 0.000955a 0.821c 0.000744a 0.800b 0.000859a 0.847 0.000568c 0.896

AMERICAN EXPRESS 0.000363 1.186 0.000372 1.217 0.000174 1.599a 0.000217 1.126

AT&T -9.59E-05 0.768b -0.000249 0.769a -0.00017 0.838 -0.000295 0.712a

BETHLEHEM STEEL 0.000434 0.796b 0.000317 0.802c 0.00062 0.748a 0.000272 0.772b

BOEING 0.000299 0.862 0.000147 0.863 0.000399 0.901 0.000239 0.932

CATERPILLAR 0.000965a 0.811c 0.000704b 0.792c 0.000943a 0.805c 0.000753b 0.857

CHEVRON - TEXACO 0.000401 0.657a 0.000296 0.585a 0.000376 0.652a 0.000227 0.601a

CITIGROUP 0.000948c 1.614a 0.000874 1.637a 0.000930c 1.540a 0.000675 1.579a

COCA - COLA 0.000658b 0.816 0.000342 0.756b 0.000406 0.756b 0.000315 0.804

E.I. DU PONT DE NEMOURS 0.000281 0.875 9.73E-05 0.904 0.00029 0.905 7.14E-05 0.861

EASTMAN KODAK -6.32E-05 0.718a -0.000183 0.699a -0.000252 0.694a -0.000246 0.692a

EXXON MOBIL 8.58E-05 1.037 8.49E-05 1.029 0.000113 0.731a 1.34E-05 0.737a

GENERAL ELECTRIC 0.000487b 0.957 0.00032 0.972 0.000276 1.018 0.000174 0.814

GENERAL MOTORS -6.13E-05 1.039 -0.000171 1.115b 1.68E-05 1.136a -6.19E-05 1.182a

GOODYEAR TIRE 0.0003 0.869 0.000152 0.844 0.000630b 0.695a 0.000414 0.701a

HEWLETT - PACKARD 0.000191 1.203 0.000131 1.179 0.000383 1.097 0.000315 1.097

HOME DEPOT 0.001074b 1.442a 0.000879b 1.202b 0.000792c 1.085 0.000788c 1.081

HONEYWELL INTL. 0.000681 1.011 0.0005 0.866 0.000516 1.203 0.000415 0.847

INTEL 0.000582 1.11 0.000609 1.003 0.000976b 1.016 0.001036b 0.986

INTL. BUS. MACH. -0.000115 0.776b -0.000266 0.763b 9.31E-06 0.859 -0.000109 0.825

INTERNATIONAL PAPER 0.000439 1.022 7.41E-05 1.174a 0.000252 1.198a 0.00015 1.230a

J.P. MORGAN CHASE & CO. 0.000774b 0.809c 0.000441 0.794c 0.000927a 0.754b 0.000507c 0.759b

JOHNSON & JOHNSON 0.00024 1.041 -4.52E-06 1.152c 0.000412c 0.819c 0.000248 0.756b

MCDONALDS 0.000325 0.795b 0.000239 0.637a 0.000372 0.757a 0.000235 0.629a

MERCK 0.000462c 0.827 0.000292 0.841 0.000388 0.963 0.000293 0.749b

MICROSOFT 0.000963c 1.065 0.000892 1.094 0.001068b 0.773b 0.000834c 0.872

Table 3.21 continued.

Panel A Panel B

selection criterion Mean return Sharpe ratio

costs per trade 0% 0.10% 0% 0.10%

Data set a b a b a b a b

MINNESOTA MNG. & MNFG. 0.000259 0.841 -4.50E-06 0.855 0.000168 0.915 2.04E-05 0.904

PHILIP MORRIS 0.000343 0.822 0.000254 0.815 0.000580b 0.688a 0.000580b 0.680a

PROCTER & GAMBLE 0.000123 0.982 4.05E-05 1.028 7.88E-05 0.647a -0.000105 0.658a

SBC COMMUNICATIONS 0.000413 0.682a 2.75E-05 0.701a 0.000226 0.776a 4.89E-05 0.763a

SEARS, ROEBUCK & CO. 0.000183 0.824c -9.44E-07 0.85 0.000203 0.864 0.000185 0.861

UNITED TECHNOLOGIES 0.000764a 0.897 0.000334 0.834 0.000944a 0.881 0.000334 0.780c

WAL - MART STORES 0.000574 1.307a 0.000496 0.968 0.000633c 0.887 0.000514 0.89

WALT DISNEY 0.000793b 0.723b 0.000403 0.776c 0.000882a 0.814b 0.000416 0.683a

B. Parameters of technical trading strategies

This appendix presents the values of the parameters of the technical trading strategy set applied in this chapter. Most parameter values are equal to those used by Sullivan et al. (1999). Each basic trading strategy can be extended by a %-band filter (band), time delay filter (delay), fixed holding period (fhp) and a stop-loss (sl). The total set consists of 787 different trading rules.

Moving-average rules

n	=number of days over which the price must be averaged
band	=%-band filter
delay	=number of days a signal must hold if you implement a time delay filter
fhp	=number of days a position is held, ignoring all other signals during this period
sl	=%-rise (%-fall) from a previous low (high) to liquidate a short (long) position

n	=[1, 2, 5, 10, 25, 50, 100, 200]
band	=[0.001, 0.005, 0.01, 0.025, 0.05]
delay	=[2, 3, 4]
fhp	=[5, 10, 25, 50]
sl	=[0.025, 0.05, 0.075, 0.10]

We combine the short run moving averages sma=1,2,5,10,25 with the long run moving averages lma=n:n > sma. With the 8 values of n we can construct 25 basic MA trading strategies. We extend these strategies with %-band filters, time delay filters, fixed holding period and a stop-loss. The values chosen above will give us in total:
25*(1+5+3+4+4) =425 MA strategies.

Trading range break rules

n	= length of the period to find local minima (support) and maxima (resistance)
band	=%-band filter
delay	=number of days a signal must hold if you implement a time delay filter
fhp	=number of days a position is held, ignoring all other signals during this period
sl	=%-rise (%-fall) from a previous low (high) to liquidate a short (long) position

n	=[5, 10, 15, 20, 25, 50 ,100, 150, 200, 250]
band	=[0.001, 0.005, 0.01, 0.025, 0.05]
delay	=[2, 3, 4]
fhp	=[5, 10, 25, 50]
sl	=[0.025, 0.05, 0.075, 0.10]

With the parameters and values given above we construct the following trading range break-out (TRB) strategies:

basic TRB strategies:	10*1	=10
TRB with %-band filter:	10*5	=50
TRB with time delay filter:	10*3	=30
TRB with fixed holding period:	10*4	=40
TRB with stop-loss:	10*4	=40

This will give in total 170 TRB strategies.

Filter rules

filt	= %-rise (%-fall) from a previous low (high) to generate a buy (sell) signal
delay	=number of days a signal must hold if you implement a time delay filter
fhp	=number of days a position is held, ignoring all other signals during this period

filt	=[0.005, 0.01, 0.015, 0.02, 0.025, 0.03, 0.035, 0.04, 0.045, 0.05,
	0.06, 0.07, 0.08, 0.09, 0.1, 0.12, 0.14, 0.16, 0.18, 0.2, 0.25,
	0.3, 0.4, 0.5]
delay	=[2, 3, 4]
fhp	=[5, 10, 25, 50]

With the parameters and values given above we construct the following filter rules (FR):

basic FR:	24*1	=24
FR with time delay:	24*3	=72
FR with fixed holding:	24*4	=96

This will give in total 192 filter strategies.

C. Parameters of recursive optimizing and testing procedure

This appendix presents the parameter values of the recursive optimizing and testing procedures applied in section 3.7. The two parameters are the length of the training period, TR, and the length of the testing period, Te. The following 36 combinations of training and testing periods, [Tr,Te], are used:

Train	Test
5	1
10	1
21	1
42	1
63	1
126	1
252	1
10	5
21	5
42	5
63	5
126	5
252	5
21	10
42	10
63	10
126	10
252	10

Train	Test
42	21
63	21
126	21
252	21
63	42
126	42
252	42
126	63
252	63
252	126
504	126
736	126
1008	126
1260	126
504	252
736	252
1008	252
1260	252

1: Separate ACFs of the returns are computed for each data series, but not presented here to save space. The tables are available upon request from the author.
2: The number of technical trading strategies is confined to 787 mainly because of computer power limitations.
3: A short position means borrowing an asset and selling it in the market. The proceeds can be invested against the risk-free interest rate, but dividends should be paid. At a later time the asset should be bought back in the market to redeem the loan. A trading strategy intends to buy back at a lower price than the asset is borrowed and sold for.
4: Blocks with geometric length are drawn from the original data series by first selecting at random a starting point in the original data series. With probability 1-q the block is expanded with the next data point in the original data series and with probability q the resampling is ended and a new starting point for the next block is chosen at random. The mean block length is then equal to 1/q. We follow Sullivan et al. (1999) by choosing q=0.10.
5: Results for the 0.50 and 1% costs per trade cases are not presented here to save space.
6: Computations are also done for the 0.25 and 0.50% transaction cost cases, but not presented here to save space.

Gerwin Griffioen's Technical Analysis Pages

Chapter 3 Technical Trading Rule Performance in Dow-Jones Industrial Average Listed Stocks

3.1 Introduction

3.2 Data and summary statistics

3.3 Technical trading strategies

3.4 Trading profits

3.5 Data snooping

3.6 Empirical results

3.6.1 Results for the mean return criterion

Technical trading rule performance

CAPM

Data snooping

3.6.2 Results for the Sharpe ratio criterion

Technical trading rule performance

CAPM

Data snooping

3.7 A recursive out-of-sample forecasting approach

3.8 Conclusion

Appendix

A. Tables

B. Parameters of technical trading strategies

Moving-average rules

Trading range break rules

Filter rules

C. Parameters of recursive optimizing and testing procedure

Chapter 3
Technical Trading Rule Performance in Dow-Jones Industrial Average Listed Stocks