1386 Discussion Papers Deutsches Institut für Wirtschaftsforschung The Weekend Effect: A Trading Robot and Fractional Integration Analysis Guglielmo Maria Caporale, Luis Gil-Alana, Alex Plastun and Inna Makarenko 2014 Opinions expressed in this paper are those of the author(s) and do not necessarily reflect views of the institute. IMPRESSUM © DIW Berlin, 2014 DIW Berlin German Institute for Economic Research Mohrenstr. 58 10117 Berlin Tel. +49 (30) 897 89-0 Fax +49 (30) 897 89-200 http://www.diw.de ISSN print edition 1433-0210 ISSN electronic edition 1619-4535 Papers can be downloaded free of charge from the DIW Berlin website: http://www.diw.de/discussionpapers Discussion Papers of DIW Berlin are indexed in RePEc and SSRN: http://ideas.repec.org/s/diw/diwwpp.html http://www.ssrn.com/link/DIW-Berlin-German-Inst-Econ-Res.html The Weekend Effect: A Trading Robot and Fractional Integration Analysis Guglielmo Maria Caporale* Brunel University, London, CESifo and DIW Berlin Luis Gil-Alana University of Navarra Alex Plastun Ukrainian Academy of Banking Inna Makarenko Ukrainian Academy of Banking May 2014 Abstract This paper provides some new empirical evidence on the weekend effect, one of the most recognized anomalies in financial markets. Two different methods are used: (i) a trading robot approach to examine whether or not there is such an anomaly giving rise to exploitable profit opportunities by replicating the actions of traders; (ii) a fractional integration technique for the estimation of the (fractional) integration parameter d. The results suggest that trading strategies aimed at exploiting the weekend effect can generate extra profits but only in a minority of cases in the gold and stock markets, whist they appear to be profitable in most cases in the FOREX. Further, the lowest orders of integration are generally found on Mondays, which can be seen as additional evidence for a weekend effect. Keywords: Efficient Market Hypothesis; weekend effect; trading strategy. JEL classification: G12, C63 * Corresponding author. Research Professor at DIW Berlin. Department of Economics and Finance, Brunel University, London, UB8 3PH. Email: Guglielmo-Maria.Caporale@brunel.ac.uk 1. Introduction Detecting calendar effects (anomalies) in financial markets is of interest both to traders aiming to exploit them to gain extra profits and to researchers analysing whether there is evidence of market failure and of the inadequacy of the Efficient Market Hypothesis (EMH). Several papers have tested for their presence using a variety of empirical methods. One of the most frequently studied anomalies is the weekend effect (Monday effect, day of the week effect) first discussed by French (1980), namely the tendency of financial assets to generate negative returns on Mondays. Different theories have been developed to account for its presence. In behavioural finance models it is attributed to the negative expectations of investors considering Monday the worst day of the week. Another possible explanation is that over the weekend market participants have more time to analyse price movements and as a result on Mondays a larger number of trades takes place. Alternatively, it might be due to deferred payments during the weekend, which create an extra incentive for the purchase of securities on Fridays leading to higher prices on that day. Overall, the empirical evidence is still mixed. The present study provides some new results based on two different methods: (i) a trading robot approach to examine whether or not there is such an anomaly giving rise to exploitable profit opportunities by replicating the actions of traders; (ii) a fractional integration technique for the estimation of the (fractional) integration parameter d. The remainder of the paper is structured as follows: Section 2 briefly reviews the literature on the weekend effect. Section 3 outlines the empirical methodology. Section 4 presents the empirical results. Section 5 offers some concluding remarks. 1 2. Literature Review Fields (1931) suggested that the best trading day of the week is Saturday. Another important study on the weekend effect is that by Cross (1973), who analysed the FridayMonday data for the Standard & Poor's Composite Stock Index from January 1953 to December 1970 and found an increase on Fridays and a decrease on Mondays.French (1980) extended the analysis to 1977 and also reported negative returns on Mondays. Further contributions by Gibbons and Hess (1981), Keim and Stambaugh (1984), Rogalski (1984), and Smirlock and Starks (1986) also found the positive-Friday / negative-Monday pattern. Connolly (1999) also allowed for heteroscedasticity but still detected a Monday effect from the mid- 1970s.Rystrom and Benson (1989) explained the presence of the dayof-the-week effect on the basis of the psychology of investors who believe that Monday is a “difficult” day of the week and have a more positive perception of Friday. Ariel (1990) argued against a connection between the weekend and the Monday effect. Agrawal and Tandon (1994) examined 19 equity markets around the world, and found the day-of-the week effect in most developed markets. Sias and Starks (1995) associate the weekend effect with stocks in large portfolios of institutional investors. Research conducted in Fortune (1998, 1999) shows that it has a tendency to disappear and is a phenomenon with two components: the first is the “weekend drift effect”, i.e. stock prices tend to decline over weekends but rise during the trading week; the second is the “weekend volatility effect”, i.e. the volatility of returns during weekends is less per day than that over contiguous trading days. As for the role of short-selling, Kazemi, Zhai, He and Cai (2013) and Chen and Singal (2003) explain the weekend effect as resulting from the closing of speculative positions on Fridays and the establishing of new short positions on Mondays by traders. However, the results of the study by Christophe, Ferri and Angel (2007) do not support this conclusion. Further evidence is provided by Singal and Tayal (2014) for the futures 2 market, Olson, Chou, Mossman (2011) who carry out various breakpoint and stability tests, and Racicot (2011) who uses spectral analysis. The findings from other relevant studies are summarised in Table 1. Table 1 Weekend effect: an overview of recent researches Object of analysis Type of (time period, Results Author analysis market, index) 1977-1991 market equity capitalization, institutional The weekend effect is driven Hypothesis Sias, Starks holdings, daily primarily by institutional investor testing (t-test (1995) returns and volume trading patterns and F-test) of 1500 institutional investors on the NYSE January 1980 -June The negative weekend drift appears 1998 - daily closeFortune to have disappeared although to-close data for the (1998) weekends continue to have low S&P volatility 500 January 1980 Jump January 1999 diffusion daily close-to-close model of data of the Dow stock returns The weekend drift effect is a Fortune 30, the S&P 500, financial anomaly that will (1999) the Wilshire 5000, ultimately correct itself. the Nasdaq Composite, and the Russell 2000 1885–1927 - the Dow Jones indexes Schwert Correlation The weekend effect seems to have portfolio; 1928– (2003) analysis disappeared since the 1980-s 2002 - the S&P composite portfolio July 1962 December 1999 New York Stock Descriptive (NYSE); December Speculative short sales can explain Chen, Singal and 1972 - December the weekend effect. (2003) regression 1999 - Nasdaq analysis daily returns for stocks; June 1988 December 1999 3 Nasdaq and January 1988 – 1999 NYSE monthly short interest data January 1988 to one-tailed December 2000 nonparametric (678 weeks) - the test based on 3:00 and closing the values for the S&P approximated 500 index; Hsaio, Solt normal April 1988 to distribution December 2000 (2004) аnd (669 weeks) - the parametric CREF stock, test to growth, and money examine the market account; strategies’ April 1994 market timing to December 2000 ability (332 weeks) – growth account September 2000 Christophe, Descriptive July 2001 daily Ferri, Angel and 9:30 am-4:00 pm regression (2007) data on NASDAQanalysis listed stock 1973 – 2007 - the Dow-Jones 30 Industrials, Standard and Poor's Regression 500, Standard & analysis, Olson, Chou, Poor’s Midcap Chow Mossman, 400, Standard & breakpoint (2011) Poor’s Smallcap tests, Bai600, NASDAQ Perron Tests 100, American Stock Exchange (AMEX) Composite indices Racicot Spectral 1970-1973 (2011) analysis S&P500 index January 1980 – present time, 60 market indices from 59 countries Descriptive Kazemi, (For all and Zhai, He and countries, except regression Cai (2013) US, major stock analysis index is used. For the US both the Dow Jones Index and the S&P 500 4 Presence of weekend effect in the average daily returns for many of the tested portfolios till 2000. Speculative short-selling does not explain the Monday-Friday difference in returns The weekend effect may have already gone through its entire involving identification, exploitation, decline, reversal, and disappearance. There is no significant weekend effect in U.S. small stocks after about mid 2003 Spectral analysis confirms the Monday effect. During the period from 1980 to 1994, short sales can explain the weekend effect. During the period from 1995 to 2007, the crosssectional weekend effect cannot be explained by short sales. were used) Singal and Tayal (2014) 3. Descriptive and regression analysis 1990 – 2012, eight futures: Crude oil, Heating Oil, Soybeans, Sugar, S&P 500 Index, British Pound, 10-Year Treasury Note, and Gold Evidence of the weekend effect in futures markets shows that security prices will generally be biased upwards, with greater overvaluation for more volatile securities. Unconstrained short selling is not a sufficient condition for unbiased prices Data and Methodology We use daily data for 35 US companies included in the Dow Jones index and 8 Blue-chip Russian companies. The sample period for the US and Russian stock markets covers the period from January 2005 and 2008 respectively till the end of April 2014. We also analyse the FOREX using data on the six most liquid currency pairs (EURUSD, GBPUSD, USDJPY, USDCHF, AUDUSD, USDCAD) and gold prices over the period from January 2000 and 2005 respectively till the end of April 2014. Our first (trading-bot) approach considers the weekend effect from the trader’s viewpoint, namely whether it is possible to make abnormal profits by exploiting it. Specifically, we programme a trading robot which simulates the actions of a trader according to an algorithm (trading strategy). To test it with historical data we use a MetaTrader trading platform which provides tools for replicating price dynamics and trades according to the adopted strategy. We examine two trading strategies: - Strategy 1: Sell on Friday close. Close position on Monday close. - Strategy 2: Sell on Monday open. Close position on Monday close. If a strategy results in the number of profitable trades > 50% and/or total profits from trading are > 0, then we conclude that there is a market anomaly. 5 Our second approach is based on estimating the degree of integration of the series for different days of the week. Specifically, we use the Whittle function in the frequency domain, as in following model: yt = α + β t + xt ; (1 − L) d xt = ut , (*) where yt is the observed time series; α and β are the intercept and the coefficient on the linear trend respectively, xt is assumed to be an I(d) process where d can be any real number, and ut is assumed to be weakly autocorrelated. However, instead of specifying a parametric ARMA model, we follow the non-parametric approach of Bloomfield (1973), which also produces autocorrelations decaying exponentially as in the AR case. If the estimated order of integration for a particular day, specifically Monday, is significantly different from that for the other days of the week, then it can be argued that there is evidence of a weekend effect. 4. Empirical Results Detailed results are presented in the Appendix. Table 1 summarises those for Strategy 1. Table 1a: Summary of testing results for Strategy 1 Type of a Profittrades Profittrades Profit>0, Totaltrades Profittrades Totalnetprofit market % oftotal %>50, % % US stock market Russian stock market FOREX GOLD 434 201 46% -1334 14% 26% 325 141 43% -285 0% 13% 724 357 49% 7726 50% 50% 453 210 46% -18733 0% 0% In general this strategy is unprofitable in the stock markets (both US and Russian) and in gold market but can generate profits in the FOREX. However, in the latter case, the number of profitable trades is less than 50%, and only for 3 of the 6 currencies analysed can profits be made. Overall, the EMH is not contradicted. 6 The corresponding results for Strategy 2 are presented in Table 1b. Table 1b: Summary of testing results for the Strategy 2 Profittrades Profit>0, Profittrades Type of Totalnetprofit Totaltrades Profittrades %>50% % % oftotal a market US stock market Russian stock market FOREX GOLD 405 190 47% -650 20% 34% 329 149 45% 40 13% 25% 724 358 49% 2738 33% 67% 449 224 50% 15673 0% 100% It appears that this strategy can be profitable in 3 of the 4 markets examined, especially in the FOREX and gold markets. However, the number of profitable trades is less than 50% in the stock market, specifically 34% and 25% using a single asset in the US (with only 12 out of 35 instruments generating profits) and Russian stock markets. The corresponding percentage for the FOREX is 67%, indicating the existence of a market anomaly in this case. These results imply that Strategy 2 (Sell on Monday open. Close position on Monday close) is much more profitable than Strategy 1 (Sell on Friday close. Close position on Monday close). The implication is that the weekend effect cannot be attributed to the arrival of new information during weekends, and that the appropriate formulation for the weekend effect is “Mondays tend to generate negative returns”. Given this mixed evidence, we also estimate the differencing parameter d for each day of the week under the three standard parameterisations of no deterministic terms, an intercept, and an intercept with a linear time trend. In the majority of cases, the lowest estimated value of d is found to be on Mondays (see Table B in the Appendix). The only two exceptions are the USDCHF and ALTRIA series, for which the lowest estimate corresponds to Friday and Wednesday respectively. However, this evidence is weak, since the unit root null hypothesis (d = 1) cannot be rejected in any case. The fact that the estimate of d is systematically smaller for Mondays than for the other days of the week 7 suggests abnormal behaviour on this day. An estimated value of d significantly smaller than 1 would imply that it is possible to make systematic profits on this day of the week using historical data. However, as can be seen in the Appendix, the confidence intervals are relatively wide in all cases, and therefore the unit root null hypothesis cannot be rejected for any day of the week, which implies weak support for a weekend effect. 5. Conclusions This paper examines one of the most recognized anomalies, i.e. the weekend effect, in various financial markets (US and Russian stock markets, FOREX, gold) applying two different methods to daily data. The first, the trading-bot approach, uses a trading robot to simulate the behaviour of traders according to a given algorithm (in our case trading on the weekend effect) and considering two alternative strategies. The second analyses the stochastic properties of the series on different days of the week by estimating their fractional integration parameter, testing if this value differs depending on the day of the week. The results can be summarised as follows. Strategy 1 (Sell on Friday close. Close position on Monday close) is unprofitable in most cases. The only possible “weekend effect” formulation is “negative returns on Mondays”. This is confirmed by the results for Strategy 2 (Sell on Monday open. Close position on Monday close): in this case it is possible to make profits, although the number of profitable deals is less than 50% and therefore it cannot be concluded that there is a market anomaly according to our criterion. The estimates of the fractional parameter d are lowest on Mondays in most cases, which is evidence in favour of the weekend effect, although the wide confidence intervals mean that this evidence is rather weak. Finally, exploitable profit opportunities based on the weekend effect are found mainly in the FOREX market. 8 References Agrawal, A., Tandon, K., 1994, Anomalies or Illusions? Evidence from Stock Markets in Eighteen Countries. Journal of international Money and Finance, №13, 83-106. Ariel, R., 1990, High Stock Returns Before Holidays: Existence and Evidence on Possible Causes. Journal of Finance, (December), 1611-1626. Bloomfield, P. (1973).An exponential model in the spectrum of a scalar time series, Biometrika 60, 217-226. Chen, H. and Singal, V., 2003, Role of Speculative Short Sales in Price Formation: The Case of the Weekend Effect. Journal of Finance. LVIII, 2. Christophe, S., Ferri, M. and Angel, J., 2007, Short-selling and the Weekend Effect in Stock Returns http://www.efmaefm.org/0EFMAMEETINGS/EFMA%20ANNUAL%20MEETINGS/ 2007-Vienna/Papers/0260.pdf. Connolly, R., 1989, An Examination of the Robustness of the Weekend Effect. Journal of Financial and Quantitative Analysis, 24, 2,133-169. Cross, F., 1973, The Behavior of Stock Prices on Fridays and Mondays. Financial Analysts Journal, November - December, 67-69. Fields, M., 1931, Stock Prices: A Problem in Verification. Journal of Business. October. 415-418. Fortune, P., 1998, Weekends Can Be Rough : Revisiting the Weekend Effect in Stock Prices. Federal Reserve Bank of Boston. Working Paper No. 98-6. Fortune, P., 1999, Are stock returns different over weekends? а jump diffusion analysis of the «weekend effect». New England Economic Review.3-19 9 French, K., 1980, Stock Returns and the Weekend Effect. Journal of Financial Economics. 8, 1, 55-69. Gibbons, M. and Hess, P., 1981, Day Effects and Asset Returns. Journal of Business, 54, no, 4, 579-596. Hsaio, P., Solt, M., 2004, Is the Weekend Effect Exploitable? Investment Management and Financial Innovations, 1, 53. Kazemi, H. S., Zhai, W., He, J. and Cai, J., 2013, Stock Market Volatility, Speculative Short Sellers and Weekend Effect: International Evidence. Journal of Financial Risk Management. Vol.2 , No. 3. 47-54. Keim, D. B. and R. F. Stambaugh, 1984, A Further Investigation of the Weekend Effect in Stock Returns, Journal of Finance, Vol. 39 (July), 819-835. Olson, D., Chou, N. T., & Mossman, C., 2011, Stages in the Life of the Weekend Effect http://louisville.edu/research/for-faculty-staff/reference-search/1999-references/2011business/olson-et-al-2011-stages-in-the-life-of-the-weekend-effect. Racicot, F-É., 2011, Low-frequency components and the Weekend effect revisited: Evidence from Spectral Analysis. International Journal of Finance, 2, 2-19. Rogalski, R. J., 1984, New Findings Regarding Day-of-the-Week Returns over Trading and Non-Trading Periods: A Note, Journal of Finance, Vol. 39, (December), 1603-1614. Rystrom, D.S. and Benson, E., 1989, Investor psychology and the day-of-the-week effect. Financial Analysts Journal (September/October), 75-78. Schwert, G. W., 2003, Anomalies and Market Efficiency. Handbook of the Economics of Finance. Elsevier Science B.V., Ch.5, 937-972. Sias, R. W., Starks, L. T., 1995, The day-of-the week anomaly: the role of institutional investors. Financial Analyst Journal. May – June.58-67. 10 Singal, V. and Tayal, J. (2014) Does Unconstrained Short Selling Result in Unbiased Security Prices? Evidence from the Weekend Effect in Futures Markets (May 5, 2014). Available at SSRN: http://ssrn.com/abstract=2433233 Smirlock, M. and Starks, L., 1986, Day-of-the-Week and Intraday Effects in Stock Returns, Journal of Financial Economics, Vol. 17, 197-210. 11 APPENDIX Table A1 US stock market, Strategy 1 Company Alcoa AltriaGroup American Express Company AmericanInternationalGroupInc ATT Inc BankofAmerica Boeing CaterpillarInc CISCO Coca-Cola DuPont ExxonMobilCorporation Freeport-McMoRan Copper&GoldInc Hewlett-Packard Company HomeDepotCorp HoneywellInternationalInc IntelCorporation InternationalPaperCompany Johnson&Johnson JP MorganChase KraftFoods McDonaldsCorporation MerckCoInc Microsoft MMM Company Pfizer ProcterGambleCompany QUALCOMM Inc Travelers UnitedParcelServiceInc United Technologies Corporation VerizonCommunicationsInc Wal-Mart StoresInc WaltDisney Yahoo! Inc Average Total trades Profit trades 442 444 442 444 441 409 444 408 409 445 445 445 206 177 224 205 184 201 212 185 187 184 215 200 Profit trades (% of total) 47% 40% 51% 46% 42% 49% 48% 45% 46% 41% 48% 45% 409 207 51% 3711 412 445 445 444 445 445 445 410 445 445 445 445 445 445 409 409 409 445 449 445 445 406 194 223 218 190 213 201 220 166 190 205 198 201 202 198 230 189 175 209 203 200 213 215 47% 50% 49% 43% 48% 45% 49% 40% 43% 46% 44% 45% 45% 44% 56% 46% 43% 47% 45% 45% 48% 53% 417 -755 -685 -1778 -832 -3261 2016 -2781 -5021 -3812 -1365 -2364 -1409 -3563 2824 27,8 -4776 -4521 -1059 -3445 -824 2977 9.1 1.0 -1.7 -1.5 -4.0 -1.9 -7.3 4.5 -6.8 -11.3 -8.6 -3.1 -5.3 -3.2 -8.0 6.9 0.1 -11.7 -10.2 -2.4 -7.7 -1.9 7.3 434 201 46% -1334 -3 12 Total net profit Profit per deal -379 -2518 747 -1003 -2253 1881 -2324 -5631 -1478 1009 -670 -3803 -0.9 -5.7 1.7 -2.3 -5.1 4.6 -5.2 -13.8 -3.6 2.3 -1.5 -8.5 Table A2 US stock market, Strategy 2 Company Alcoa AltriaGroup American Express Company AmericanInternationalGroupInc ATT Inc BankofAmerica Boeing CaterpillarInc CISCO Coca-Cola DuPont ExxonMobilCorporation Freeport-McMoRan Copper&GoldInc Hewlett-Packard Company HomeDepotCorp HoneywellInternationalInc IntelCorporation InternationalPaperCompany Johnson&Johnson JP MorganChase KraftFoods McDonaldsCorporation MerckCoInc Microsoft MMM Company Pfizer ProcterGambleCompany QUALCOMM Inc Travelers UnitedParcelServiceInc United Technologies Corporation VerizonCommunicationsInc Wal-Mart StoresInc WaltDisney Yahoo! Inc Average Total trades Profit trades 412 413 412 413 410 384 413 385 384 413 413 413 204 184 218 231 182 204 190 188 173 175 180 180 Profit trades (% of total) 50% 45% 53% 56% 44% 53% 46% 49% 45% 42% 44% 44% 384 202 53% 7284 383 413 413 413 413 413 413 382 413 413 413 413 413 413 384 384 384 163 197 200 187 206 180 197 174 179 181 197 172 178 173 197 185 161 43% 48% 48% 45% 50% 44% 48% 46% 43% 44% 48% 42% 43% 42% 51% 48% 42% -2305 -679 -190 -1137 61 -2377 2259 -1374 -3537 -2268 -1165 -1977 -1185 -3806 1693 320 -3972 413 201 49% -2158 416 413 413 383 207 189 208 211 50% 46% 50% 55% 140 -2782 -5 2311 405 190 47% -650 13 Total net profit Profit per deal 594 -1389 1194 1227 -1179 2840 -851 78 -1091 -2691 -594 -4024 1.4 -3.4 2.9 3.0 -2.9 7.4 -2.1 0.2 -2.8 -6.5 -1.4 -9.7 19.0 -6.0 -1.6 -0.5 -2.8 0.1 -5.8 5.5 -3.6 -8.6 -5.5 -2.8 -4.8 -2.9 -9.2 4.4 0.8 -10.3 -5.2 0.3 -6.7 0.0 6.0 -2 Table A3 Russian stock market, Strategy 1 Company GAZPROM NORILSKY NICKEL LUKOIL ROSNEFT SBERBANK GAZPROM NEFT SURGUTNEFTEGAZ VTB BANK Average Total trades Profit trades Profit trades (% of total) Total net profit Profit per deal 335 373 393 218 365 357 240 315 325 153 174 190 98 158 143 103 111 141 46% 47% 48% 45% 43% 40% 43% 35% 43% -81 -1540 1857 -117 -1262 -228 -540 -369 -285 -0.2 -4.1 4.7 -0.5 -3.5 -0.6 -2.3 -1.2 -0.96 Table A4 Russian stock market, Strategy 2 Company GAZPROM NORILSKY NICKEL LUKOIL ROSNEFT SBERBANK GAZPROM NEFT SURGUTNEFTEGAZ VTB BANK Average Total trades Profit trades Profit trades (% of total) Total net profit Profit per deal 325 359 376 210 352 345 359 306 329 135 180 186 89 171 141 168 120 149 42% 50% 49% 42% 49% 41% 47% 39% 45% -345 1055 1295 -200 -257 -321 -657 -254 40 -1.1 2.9 3.4 -1.0 -0.7 -0.9 -1.8 -0.8 0.01 14 Table A5 FOREX, Strategy 1 Asset EURUSD GBPUSD USDCHF USDJPY AUDUSD USDCAD Average Total trades Profit trades Profit trades (% of total) Total net profit Profit per deal 724 724 724 724 724 724 724 367 364 334 370 358 349 357 51% 50% 46% 51% 49% 48% 49% 25948 48839 -17523 9807 -4671 -16044 7726 36 67 -24 14 -6 -22 11 TABLE A6 FOREX, Strategy 2 Asset EURUSD GBPUSD USDCHF USDJPY AUDUSD USDCAD Average Total trades Profit trades Profit trades (% of total) Total net profit Profit per deal 724 724 724 724 724 724 724 363 360 355 377 337 357 358 50% 50% 49% 52% 47% 49% 49% 18640 20576 -16479 6281 554 -13142 2738 26 28 -23 9 1 -18 4 15 TABLE A7 Gold, Strategy 1 Asset Gold Total trades Profit trades Profit trades (% of total) Total net profit Profit per deal 453 210 46% -18733 -41 TABLE A8 Gold, Strategy 2 Asset Gold Total trades Profit trades Profit trades (% of total) Total net profit Profit per deal 449 224 50% 15673 35 16 Estimates of d in a model with autocorrelated errors Table B1: Estimates of d in a model with autocorrelated errors: GOLD Day of the week No regressors An intercept A linear time trend Monday 0.930 (0.855, 1.064) 0.939 (0.866, 1.032) 0.939 (0.865, 1.035) Tuesday 0.930 (0.854, 1.047) 0.942 (0.871, 1.044) 0.942 (0.877, 1.042) Wednesday 0.938 (0.841, 1.064) 0.949 (0.872, 1.062) 0.950 (0.876, 1.068) Thursday 0.937 (0.843, 1.055) 0.946 (0.866, 1.053) 0.946 (0.864, 1.057) Friday 0.936 (0.840, 1.060) 0.943 (0.865, 1.054) 0.943 (0.863, 1.057) Table B2: Estimates of d in a model with autocorrelated errors: EURUSD Day of the week No regressors An intercept A linear time trend Monday 0.954 (0.877, 1.044) 0.963 (0.885, 1.066) 0.963 (0.885, 1.063) Tuesday 0.958 (0.884, 1.037) 0.991 (0.900, 1.092) 0.992 (0.902, 1.092) Wednesday 0.961 (0.886, 1.055) 1.010 (0.921, 1.107) 1.010 (0.924, 1.107) Thursday 0.964 (0.876, 1.045) 1.008 (0.936, 1.106) 1.008 (0.935, 1.106) Friday 0.972 (0.890, 1.050) 1.003 (0.914, 1.104) 1.003 (0.914, 1.098) Table B3: Estimates of d in a model with autocorrelated errors: USDCHF Day of the week No regressors An intercept A linear time trend Monday 1.008 (0.940, 1.104) 0.936 (0.856, 1.042) 0.936 (0.856, 1.045) Tuesday 1.016 (0.945, 1.117) 0.937 (0.857, 1.044) 0.936 (0.857, 1.042) Wednesday 1.012 (0.941, 1.113) 0.929 (0.853, 1.030) 0.929 (0.842, 1.032) Thursday 1.015 (0.931, 1.098) 0.930 (0.843, 1.013) 0.930 (0.846, 1.012) Friday 1.002 (0.920, 1.089) 0.928 (0.850, 1.034) 0.928 (0.843, 1.034) 17 Table B4: Estimates of d in a model with autocorrelated errors: LUKOIL Day of the week No regressors An intercept A linear time trend Monday 0.987 (0.888, 1.118) 0.858 (0.736, 1.035) 0.858 (0.734, 1.035) Tuesday 0.989 (0.882, 1.155) 0.859 (0.739, 0.978) 0.859 (0.739, 0.977) Wednesday 0.934 (0.837, 1.059) 0.868 (0.752, 1.024) 0.868 (0.752, 1.019) Thursday 1.007 (0.883, 1.143) 0.927 (0.793, 1.073) 0.921 (0.802, 1.075) Friday 1.002 (0.905, 1.136) 0.898 (0.767, 1.057) 0.898 (0.776, 1.055) Table B5: Estimates of d in a model with autocorrelated errors: GAZPROM Day of the week No regressors An intercept A linear time trend Monday 0.939 (0.820, 1.184) 0.963 (0.836, 1.102) 0.963 (0.836, 1.102) Tuesday 0.962 (0.845, 1.107) 0.992 (0.857, 1.144) 0.992 (0.855, 1.142) Wednesday 0.954 (0.841, 1.100) 0.982 (0.863, 1.130) 0.982 (0.863, 1.132) Thursday 0.962 (0.831, 1.118) 0.997 (0.863, 1.155) 0.997 (0.862, 1.155) Friday 0.939 (0.877, 1.089) 0.987 (0.860, 1.131) 0.988 (0.861, 1.132) Table B6: Estimates of d in a model with autocorrelated errors: ALTRIA Day of the week No regressors An intercept A linear time trend Monday 1.005 (0.910, 1.122) 1.008 (0.916, 1.132) 1.007 (0.915, 1.133) Tuesday 0.993 (0.925, 1.097) 0.992 (0.907, 1.094) 0.992 (0.907, 1.096) Wednesday 0.986 (0.911, 1.090) 0.971 (0.883, 1.076) 0.971 (0.883, 1.076) Thursday 0.986 (0.913, 1.103) 0.979 (0.903, 1.085) 0.979 (0.903, 1.086) Friday 1.001 (0.917, 1.093) 0.991 (0.900, 1.091) 0.994 (0.900, 1.091) Table B7: Estimates of d in a model with autocorrelated errors: FREEPORT Day of the week No regressors An intercept A linear time trend Monday 1.042 (0.944, 1.183) 1.047 (0.944, 1.183) 1.047 (0.943, 1.190) Tuesday 1.096 (0.984, 1.232) 1.050 (0.990, 1.255) 1.064 (0.990, 1.255) Wednesday 1.074 (0.960, 1.210) 1.073 (0.962, 1.204) 1.072 (0.960, 1.204) Thursday 1.044 (0.943, 1.199) 1.044 (0.943, 1.179) 1.049 (0.943, 1.179) Friday 1.067 (0.967, 1.221) 1.088 (0.962, 1.224) 1.088 (0.962, 1.225) 18