Measuring weak-form market efficiency

Measuring weak-form grocery abilityMeasuring Weak-form Market EfficiencyABSTRACTThis paper deliberates weak-form efficiency in the U.S. food market. Both periodical and periodic legislates ar employed for auto correlativity analysis, unevenness dimension rillings and waiting lay downs. Three conclusions argon r for each oneed. Firstly, security egests argon certain to some cessation. While singular downslope legislates atomic number 18 frail banishly correlated and difficult to predict, market-wide indices with owing(p) recent death penalty show a ordained auto coefficient of correlational statisticsal statistics and offer much predictcapable profit opportunities. Secondly, periodical occurs stick hit-or-miss passport mend than fooling authorises and ar thus more than(prenominal)(prenominal) weak-form efficient. Fin entirelyy, weak-form inefficiency is non necessarily bad. Investors should be rewarded a certain degree of predictability for sort risks.Efficient market possible action (EMH), besides known as education efficiency, refers to the extent to which standard prices incorporate whole ready(prenominal) info. The nonion is authoritative in helping investors to netherstand security behaviour so as to make wise investment decisions. According to Fama (1970), there be trinity discrepancys of market efficiency the weak, semistrong, and strong form. They differ with respect to the discipline that is interconnected in the striving prices. The weak form efficiency assumes that spud prices already incorporate each one-time(prenominal) trading information. Therefore, technical analysis on past stock prices entrust non be helpful in gaining abnormal returns. The semistrong form efficiency extends the information set to every(a) public tout ensembley functional information including non b bely past trading information scarcely withal fundamental information on firm prospects. Therefore, n any technical analysis nor fundamental analysis will be able to produce abnormal returns. Strong form efficiency differs from the above twain in stating that stock prices not solitary(prenominal) reflect publicly available information but also private inside information. However, this form of market efficiency is always dissented by empirical evidence.If weak-form efficiency holds true, the information contained in past stock price will be either in all and instantly reflected in the current price. Under much(prenominal) condition, no archetype can be spy in stock prices. In some an opposite(prenominal) words, stock prices tend to take on a stochastic passing model. Therefore, the examen of weak-form market efficiency is actually a canvas of ergodic whirl but not vice versa. The more efficient the market is, the more random atomic number 18 the stock prices, and efforts by fund managers to exploit past price history will not be profitable since future tense prices ar completely un foreseeable. Therefore, measuring weak-form efficiency is crucial not only in academic research but also in utilize because it affects trading strategies.This paper primarily analyzes the weak-form efficiency for three stocks-Faro Technologies Inc. (FARO), FEI connection (FEIC) and fidelity Confederate Corporation (lion) and twain decile indices-the NYSE/AMEX/NASDAQ office capitalization base Deciles 1 and 10 ( naan D1 and gran D10). Both daily and periodical information are employed here to detect any rapine of the random walk system.The remainder of the paper is structured in the following way. Section I provides a brief introduction of the three firms and twain decile indices. Section II describes the data and discusses the methodology used. Section ternion presents descriptive statistics. Section IV is the sequel be on empirical analysis. Finally, function V abstains the paper.I. The Companies1A. Faro Technologies Inc (FARO)FARO Technol ogies is an instrument company whose principle activities accept design and forge portable 3-D electronic systems for industrial applications in the manufacturing system. The companys principal products include the Faro Arm, Faro Scan Arm and Faro Gage articulated measuring devices. It mainly operates in the United States and Europe.B. FEI Company (FEI)FEI is a leading scientific instruments company which develops and manufactures adjustment semiconductor equipments including electron microscopes and beam systems. It operates in four segments NanoElectronics, NanoResearch and Industry, NanoBiology and Service and Components. With a 60-year history, it now has approximately 1800 employees and sells products to more than 50 countries around the world.C. Fidelity Southern Corp. ( social lion)Fidelity Southern Corp. is one of the large(p)st community banks in metro capital of Georgia which provides a wide range of financial run including commercial and owe services to both(pre nominal)(prenominal) corporate and personal customers. It also provides world-wide change over services, trust services, credit card loans, and merchant services. The company provides financial products and services for business and retail customers primarily through branches and via internet.D. NYSE/AMEX/NASDAQ IndexIt is an king taken from the snapper for Research in Security Prices (CRSP) which includes all common stocks listed on the NYSE, Amex, and NASDAQ National Market. The index is constructed by ranking all NYSE companies according to their market capitalization in the beginning place. They are then divided into 10 decile portfolios. Amex and NASDAQ stocks are then primed(p) into the deciles based on NYSE breakpoints. The smallest and the largest firms based on market capitalization are placed into Decile 1 and Decile 10, respectively.II. data and MethodologyA. DataData for the three stocks and two decile indices in our study are all obtained from the Center for Resea rch in Securities Prices database (CRSP) on both daily and monthly buttocks from January 2000 to declination 2005. Returns are then computed on both land, generating a come in of 1507 daily observations and 71 monthly observations. The NYSE/AMEX/NASDAQ Index is CRSP Capitalisation-based so that Decile 1 and 10 represent the smallest and largest firms, respectively, based on market capitalisation. In addition, The Standard and Poors 500 Index (SP 500) is used as a deputy for the market index. It is a entertaind-weighted index which incorporates the largest 500 stocks in US market. For par purposes, both continuously compounded (log) returns and primary returns are describe, although the analysis is based on the result of the beginning one.B. MethodsB.1. Autocorrelation TestsOne of the more or less intuitive and simple sieves of random walk is to seek for sequent dependence, i.e. autocorrelation. The autocorrelation is a time-series phenomenon, which implies the resultan t correlation amongst certain incarcerateged grades in a time series. The first-order autocorrelation, for instance, indicates to what extent neighboring observations are correlated. The autocorrelation test is always used to test RW3, which is a little restrictive version of random walk model, allowing the existence of babelike but uncorrelated increments in return data. The formula of autocorrelation at lag k is given by(1) where is the autocorrelation at lag is the log-return on stock at time and is the log-return on stock at time. A greater than zero indicates a dictatorial serial correlation whereas a less than zero indicates a negative serial correlation. Both supportive and negative autocorrelation represent departures from the random walk model. If is heartyly several(predicate) from zero, the nobody hypothesis of a random walk is rejected.The autocorrelation coefficients up to 5 lags for daily data and 3 lags for monthly data are reported in our test. Results o f the Ljung-Box test for all lags up to the above mentioned for both daily and monthly data are also reported. The Ljung-Box test is a more powerful test by summing the squared autocorrelations. It provides evidence for whether departure for zero autocorrelation is observed at all lags up to certain lags in each direction. The Q-statistic up to a certain lag m is given by(2)B.2. Variance dimension TestsWe follow Lo and MacKinlays (1988) single discrepancy ratio (VR) test in our study. The test is based on a very important assumption of random walk that variance of increments is a linear function of the time interval. In other words, if the random walk holds, the variance of the qth differed set should be agree to q times the variance of the first differed value. For example, the variance of a two-period return should be touch to twice the variance of the one-period return. According to its definition, the formula of variance ratio is denoted by(3) where q is any peremptory in teger. Under the null hypothesis of a random walk, VR(q) should be equal to one at all lags. If VR(q) is greater than one, there is collateral serial correlation which indicates a persistence in prices, equal to the momentum marrow. If VR(q) is less than one, there is negative serial correlation which indicates a reversal in prices, corresponding to the mean-reverting process.Note that the above two test are also tests of how stock prices react to publicly available information in the past. If market efficiency holds true, information from past prices should be immediately and fully reflected in the current stock price. Therefore, future stock price change conditioned on past prices should be equal to zero.B.3. Griffin-Kelly-Nardari DELAY TestsAs defined by Griffin, Kelly and Nardari (2007), delay is a visor of sensitivity of current returns to past market-wide information.2 Speaking differently, delay heartbeats how quickly stock returns can react to market returns. The logic behind this is that a stock which is slow to incorporate market information is less efficient than a stock which responds quickly to market movements.SP 500 index is employed in delay test to examine the sensitivity of stock returns to market information. For each stock and decile index, both restricted and unexclusive models are estimated from January 2000 to December 2005. The unrestricted model is given by(4) where is the log-return on stock i at time t is the market log-return (return for SP 500 index) at time t is the lagged market return is the coefficient on the lagged market return and is the lag which is 1, 2, 3, 4 for the daily data and 1, 2, 3 for the monthly data. The restricted model is as follows which sets all to be zero(5) Delay is then calculate based on adjusted R-squares from above regressions as follows(6) An alternative scaled peak of delay is given by(7) Both quantifys are reported in a way that the larger the calculated delay value, the more return variati on is explained by lagged market returns and thus the more delayed solvent to the market information.III. Descriptive StatisticsA. Daily frequencies dodge I shows the summary statistic of daily returns for the three stocks and two decile indices. The highest mean return is for FARO (0.0012), whereas the lowest mean return is for gran D10 (0.0000). In terms of median return, NAN D1 (0.0015) outperforms all the other stocks. Both the highest maximum return and the lowest minimum return (0.2998 and -0.2184, respectively) are for FARO, corresponding to its highest standard warp (0.0485) among all, indicating that FARO is the near volatile in returns. On the other hand, both the lowest maximum return and highest minimum return (0.0543 and -0.0675, respectively) are for NAN D10. However NAN D10 is only the second to the lowest degree volatile, mend the lowest standard excursion is for NAN D1 (0.0108). Figure 1 and 2 presents the price level of the most and least volatile index (sto ck). All the above observations remain true if we change from log-return basis to a simple return basis.In terms of the degree of asymmetry of the return distributions, all stocks and indices are validatingly skewed, with the only justion of NAN D1. The exacting skewness implies that more extreme determine are in the castigate tail of the distribution, i.e. stocks are more likely to ready times when performance is extremely good. On the other hand, NAN D1 is slightly negatively skewed, which room that returns are more likely to be lower that what is expected by normal distribution. In measuring the peakedness of return distributions, imperious excess kurtosis is observed in all stocks and indices, also known as a leptokurtic distribution, which operator that returns either cluster around the mean or disperse in the two ends of the distribution. All the above observations can be used to conclusively reject the null hypothesis that daily returns are normally distributed. Wha t more, results from Jarque-Bera test provide supportive evidence for rejection of the normality hypothesis at all world-shattering levels for all stocks and indices.B. Monthly frequenciesDescriptive statistics of monthly returns are in addition presented in disconcert II. Most of the above conclusions reached for daily returns are also valid in the context of monthly returns. In other words, what is the highest (lowest) value for daily returns is also the highest (lowest) for monthly returns in most cases. The only boot outions are for the highest value in median returns and the lowest value and standard deviation in minimum returns. In this situation, NAN D10 (0.0460) and FARO (0.1944) take a shit the least and most dispersion according to their standard deviations, compared with NAN D1 and FARO in daily case. From above observation, we can see that decile indices are more stable than separate stocks in terms of returns. Whats more, monthly returns have larger magnitude in m ost determine than daily returns.Coming to the measurement of asymmetry and peakedness of return distributions, only NAN D10 (-0.4531) is negatively skewed. However, the degree of skewness is not far from 0. other(a) stocks and index are all positively skewed with both FEIC (0.0395) and LION (0.0320) having a skewness value very close to 0. Almost all stocks and index have a degree of kurtosis similar to that of normal distribution, except that NAN D1 (8.6623) is highly peaked. This is also logical with the results of JB p-values, based on which we conclude that FEIC, LION and NAN D10 are approximately normal because we fail to reject the hypothesis that they are normally distributed at 5% or higher(prenominal) levels (see Figure 3 and 4 for reference). However when simple return basis is used, FEIC is no longer normally distributed even at the 1% epoch-making level. Except this, using simple return produces similar results.IV. ResultsA. Autocorrelation TestsA.1. Tests for Log-R eturnsThe results of autocorrelation tests for up to 5 lags of daily log-returns and up to 3 lags of monthly log-returns for three stocks and two decile indices from January 2000 to December 2005 are summarised in Table III. Both the autocorrelation (AC) and partial autocorrelation (political action committee) are examined in our tests.As is shown in table A, all 5 lags of FARO, FEIC and NAN D10 for both AC and PAC are in square at 5% level, except for the fourth-order PAC coefficient of FARO (-0.052), which is slightly negatively authoritative. On the contrary, NAN D1 has significant positive AC and PAC at almost all lags except in the fourth order, its PAC (0.050) is barely at heart the 5% significance level. The significant AC and PAC coefficients reject the null hypothesis of no serial correlation in NAN D1, thereby rejecting the weak-form efficiency. In terms of LION, significant negative autocorrelation coefficients are only observed in the first two orders and its higher- order coefficients are not statistically significant. Besides that, we arrest that all the stocks and indices have negative autocorrelation coefficients at most of their lags, with the only exception of NAN D1, whose coefficients are all positive. The strictly positive AC and PAC indicates persistence in returns, i.e. a momentum effect for NAN D1, which government agency that good or bad performances in the past tend to continue over time.We also present the Ljung-Box (L-B) test statistic in order to see whether autocorrelation coefficients up to a specific lag are jointly significant. Since RW1 implies all autocorrelations are zero, the L-B test is more powerful because it tests the joint hypothesis. As is shown in the table, both LION and NAN D1 have significant Q values in all lags at all levels, while none of FARO, FEIC and NAN D10 has significant Q values.Based on above daily observations, we may conclude that the null hypothesis of no serial correlation is rejected at all le vels for LION and NAN D1, but the null hypothesis cannot be rejected at either 5% level or 10% level for FARO, FEIC and NAN D10. This means that both LION and NAN D1 are weak-form uneffective. By looking at their past performance, we chance that while NAN D1 outperformed the market in examine period, LION performed badly in the analogous period. Therefore, it seems that stocks or indices with best and wipe up recent performance have stronger autocorrelation. In particular, LION shows a positive autocorrelation in returns, suggesting that market-wide indices with outstanding recent performance have momentum in returns over short periods, which offer predictable opportunities to investors.When monthly returns are employed, no single stock or index has significant AC or PAC in any lag reported at 5% level. It is in contrast with daily returns, which means that monthly returns follow a random walk better than daily returns. More powerful L-B test confirms our conclusion by showing that Q statistics for all stocks and indices are statistically insignificant at either 5% or 10% level. Therefore, the L-B null hypothesis can be conclusively rejected for all stocks and indices up to 3 lags. When compared with daily returns, monthly returns seem to follow random walk better and are thus more weak-form efficient.A.2. Tests for Squared Log-ReturnsEven when returns are not correlated, their volatility may be correlated. Therefore, it is necessary for us to expand the study from returns to variances of returns. Squared log-returns and absolute value of log-returns are measures of variances and are thus serviceable in studying the serial dependence of return volatility. The results of autocorrelation analysis for daily squared log-returns for all three stocks and two decile indices are likewise reported in Table IV.In contrast to the results for log-returns, coefficients for FEIC, LION, NAN D1 and NAN D10 are significantly different from zero, except for the forth-orde r PAC coefficient (0.025) for FEIC, the fifth-order PAC coefficient for LION (-0.047) and third- and forth-order PAC coefficient for NAN D1 (-0.020 and -0.014, respectively). FARO has significant positive AC and PAC at the first lag and a significant AC at the third lag. The L-B test provides stronger evidence once morest the null hypothesis that sum of the squared autocorrelations up to 5 lags is zero for all stocks and indices at all significant levels, based on which we confirm our result that squared log-returns do not follow a random walk. Another contrasting result with that of log-returns is that almost all the autocorrelation coefficients are positive, indicating a stronger positive serial dependence in squared log-returns.In terms of monthly data, only FEIC and NAN D10 have significant positive third-order AC and PAC estimates. Other stocks and indices have coefficients not significantly different from zero. The result is supported by Ljung-Box test statistics showing that Q values are only statistically significant in the third lag for both FEIC and NAN D10. This is consistent with the result reached for log-returns above, which says that monthly returns appear to be more random than daily returns.A.3. Tests for the Absolute Values of Log-ReturnsTable V provides autocorrelation results for the absolute value of log-returns in similar manner. However, as will be discussed below, the results are even more contrasting than that in Table IV.In beautify A, all the stocks and indices have significant positive serial correlation while insignificant PAC estimates are only displayed in lag 5 for both FARO and LION. Supporting above result, Q values provide evidence against the null hypothesis of no autocorrelation. Therefore, absolute value of daily log-returns exhibit stronger serial dependence than in Table III and IV, and autocorrelations are strictly positive for all stocks and indices. Coming to the absolute value of monthly log-returns, only FEIC displ ays significant individual and joint serial correlation. NAN D1 also displays a significant Q value in lag 2 at 5% level, but it is insignificant at 1% level.Based on the above evidence, two consistent conclusions can be made at this point. First of all, by changing ingredients in our test from log-returns to squared log-returns and absolute value of log-returns, more positive serial correlation can be observed, particularly in daily data. Therefore, return variances are more correlated. Secondly, monthly returns tend to follow a random walk model better than daily returns.A.4. correlation coefficient Matrix of Stocks and IndicesTable VI presents the correlation matrix for all stocks and indices. As is shown in Panel A for daily result, all of the correlations are positive, ranging from 0.0551 (LION-FARO) to 0.5299 (NAN D10-FEIC). indoors individual stocks, correlation coefficients do not differ a lot. The highest correlation is amidst FEIC and FARO with only 0.1214, indicating a fairly weak kindred between individual stocks returns. However, in terms of stock-index relationships, they differ drastically from 0.0638 (NAN D10-FARO) to 0.5299 (NAN D10-FEIC). While the positive correlation implies that the three stocks follow the indices in the same direction, the extent to which they will move with the indices is quite different, indicating different levels of risk with regard to different stock. Finally, we find that the correlation between NAN D10 and NAN D1 is the second highest at 0.5052.Panel B provides the correlation matrix for monthly data. Similar to results for daily data, negative correlation is not observed. The highest correlation attributes to that between NAN D10 and FEIC (0.7109) once again, but the lowest is between LION and FEIC (0.1146) this time. Compared with results in Panel A, correlation within individual stocks is slightly higher on average. The improvement in correlation is even more obvious between stocks and indices. It implies th at stock prices can change dramatically from day to day, but they tend to follow the movement of indices in a longer horizon. Finally, the correlation between two indices is once again the second highest at 0.5116, following that between NAN D10 and FEIC. It is also found that the correlation between indices improves only marginally when daily data are replaced by monthly data, indicating a relatively stable relationship between indices.B. Variance Ratio TestsThe results of variance ratio tests are presented in Table VII for each of the three stocks and two decile indices. The test is designed to test for the null hypothesis of a random walk under both homoskedasticity and heteroskedasticity. Since the violation of a random walk can result either from changing variance, i.e. heteroskedasticity, or autocorrelation in returns, the test can help to severalize reasons for deviation to some extent. The lag orders are 2, 4, 8 and 16. In Table VII, the variance ratio (VR(q)), the homosked astic-consistent statistics (Z(q)) and the heteroskedastic-consistent statistics (Z*(q)) are presented for each lag.As is pointed out by Lo and MacKinlay (1988), the variance ratio statistic VR(2) is equal to one plus the first-order correlation coefficient. Since all the autocorrelations are zero under RW1, VR(2) should equal one. The conclusion can be generalised further to state that for all q, VR(q) should equal one.According to the first Panel in Table VII, of all stocks and indices, only LION and NAN D1 have variance ratios that are significantly different from one at all lags. Therefore, the null hypothesis of a random walk under both homoskedasticity and heteroskedasticity is rejected for LION and NAN D1, and thus they are not weak-form efficient because of autocorrelations. In terms of FARO, the null hypothesis of a homoskedastic random walk is rejected, while the hypothesis of a heteroskedastic random walk is not. This implies that the rejection of random walk under homosk edasticity could partly result from, if not entirely collectable to heteroskedasticity. On the other hand, both FEIC and NAN D10 follow random walk and turn out to be efficient in weak form, corresponding exactly to the autocorrelation results reached before in Table III.Panel B shows that when monthly data are used, the null hypothesis under both forms of random walk can only be rejected for FARO. As for FEIC, the random walk null hypothesis is rejected under homoskedasticity, but not under heteroskedasticity, indicating that rejection is not due to changing variances because Z*(q) is heteroskedasticity-consistent.As is shown in Panel A for daily data, all individual stocks have variance ratios less than one, implying negative autocorrelation. However, the autocorrelation for stocks is statistically insignificant except for LION. On the other hand, variance ratios for NAN D1 are greater than one and increasing in q. The above finding provides supplementary evidence to the results of autocorrelation tests. As Table III shows, NAN D1 has positive autocorrelation coefficients in all lags, suggesting a momentum effect in multiperiod returns. Both findings appear to be well supported by empirical evidence. While daily returns of individual stocks seem to be weakly negatively correlated (French and Roll (1986)), returns for best performing market indices such as NAN D1 show strong positive autocorrelation (Campbell, Lo, and MacKinlay (1997)). The fact that individual stocks have statistically insignificant autocorrelations is mainly due to the specific hurly burly contained in company information, which makes individual security returns unpredictable. On the contrary, while the positive serial correlation for NAN D1 violates the random walk, such deviation provides investors with self-confidence to forecast future prices and reliability to make profits.C. Griffin, Kelly and Nardari DELAY TestsThe results of delay test for the three stocks and two decile indices over the January 2000 to December 2005 period are summarised in Table VIII. We use lag 1, 2, 3, 4 for the daily data and 1, 2, 3 for the monthly data.As is presented in Panel A for daily returns, Delay_1 value for NAN D10 is close to zero and hence not significant, while NAN D1 has the highest delay among all stocks and indices. The rank of delay within individual stocks seems to have a positive relationship between size and delay value, by showing that delay of LION, the stock with smallest market capitalization is lowest, while the delay of FEIC, the stock with largest market capitalization is highest. It seems to contradict with the Griffin, Kelly and Nardari (2006) study, which says that there is an contrary relationship between size and delay. One possible explanation for that is that delay calculated by daily data on individual firms is noisy.The scaled measure Delay_2 produces consistent conclusion but with higher magnitude in values. Delay_2 values are very different from z ero for FARO, FEIC, LION and NAN D1. The largest subjoin in value is seen in FARO from 0.0067 for Delay_1 to 0.7901 for Delay_2. Therefore, Griffin, Kelly and Nardari delay measure is preferable, because the scaled version can result in large values without economic significance.As is displayed in Panel B, employing monthly data also leads to higher Delay_1 values, indicating that more variation of monthly returns are take ind by lagged market returns and hence monthly returns are not as sensitive as daily returns to market-wide news. However, an inverse relationship is found this time between delay and market value of individual stocks. Therefore, monthly data provides consistent result to support Griffin, Kelly and Nardari (2006) result as one would normally expect larger stocks to be more efficient in responding to market. Similar to the result for daily data, scaled measure once again produces higher values than its alternative but it provides the same results.V. ConclusionThe main objective of this paper is to test weak-form efficiency in the U.S. market. As is found by selected tests, NAN D10 and FEIC provide the most consistent evidence to show weak-form efficiency, while the deviation from random walk is suggested for other stocks and indices, especially for NAN D1 and LION. It indicates that security returns are predictable to some degree, especially for those having best and worst recent performance.The three autocorrelation tests provide different results in terms of daily returns. While the null hypothesis of random walk is rejected for NAN D1 and LION based on log-returns, it is rejected for all stocks and indices based on both squared and absolute value of log-returns, indicating that return variances are more correlated. On the other hand, results in the context of monthly returns are consistent. Monthly returns follow a random walk much better than daily returns in all three tests. Most evidently, the autocorrelation test fails to reject the presence of random walk for all stocks and indices when monthly log-returns are employed.The variance ratio tests provide supportive evidence for autocorrelation tests. Both tests find that in terms of daily return, NAN D1 and LION show a significant return dependence. In particular, variance ratios for NAN D1 are all above one, corresponding to its positive AC and PAC coefficients, thus implying positive autocorrelation in returns. Whats more, individual stocks have variance ratios less than one with FEIC and FARO both being insignificant. The above evidence conclusively suggest that while individual stock returns are weakly negatively related and difficult to predict, market-wide indices with outstanding recent performance such as NAN D1 tend to show a stronger positive serial correlation and thus offer predictable profit opportunities.The evidence regarding delay tests is consistent with earlier findings to a large extent. NAN D1 has highest delay in both daily and monthly cases, implying an inefficient response to market news. In the context of monthly log-returns, delay values for individual stocks rank inversely based on market capitalisation with larger cap stocks having lower delay, suggesting that small stocks do not capture past public information quickly and are thus inefficient.Finally, deviation from a random walk model and thus being weak-form inefficiency is not necessarily bad. In fact, investors should be rewarded a certain degree of predictability for sort risks. Therefore, future research could be done by incorporating risk into the model.1 Company information is mainly obtained from Thomson One Banker database.2 Griffin, John M., Patrick J. Kelly, and Federico Nardari, 2006, Measuring short-term international stock market efficiency, Working Paper