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CHAPTER 5 Technical Analysis And Weak Form Market Efficiency

A. Technical Analysis for Stock

Technical analysis is concerned with the examination of historical market price

and volume sequences to evaluate or time securities transactions. Technical analysis is based on the concept that all information regarding securities, including earnings, risk, products, etc. is reflected in market behavior. Market price sequences are of primary importance to the buy or sell decision; many technical analysts focus on charting historical market prices of securities. In his best-selling book on technical analysis, Edwards and Magee [1997] argue The market price reflects not only the differing fears and guesses and moods, rational and irrational, of hundreds of potential buyers and sellers, but it also reflects their needs and resources in total, factors which defy analysis and for which no statistics are available. The market price and its behavior over time provide meaningful information. Hence, the current price may not be the best indicator of the intrinsic value of a stock; in fact, it may well be futile to attempt to determine this intrinsic value. In addition to price histories, which indicate the psychology of the market better than firm fundamental factors, market volume and information regarding other participants in markets will probably be important to most technical analysts.

The theoretical foundation for technical analysis is derived from the following set of assumptions:

1. Market value is determined by the interaction of supply and demand,

which are functions of a variety of factors, both rational and irrational. 2. Security prices tend to move in trends that persist for an appreciable length

of time, despite minor fluctuations in the market. Many of these trends will repeat over time in a rather consistent manner.

3. Changes in a trend are caused by the shifts in supply and demand. 4. Shifts in supply and demand, no matter why they occur, can be detected

sooner or later in charts or sequences of market transactions. Many technical analysts (chartists) assume that security prices and market

behavior are based on the "psychology" of the market. This psychology is revealed through historical price sequences and charts. A given sequence or pattern may be associated with a given market psychology, which will result in the same future price outcome as did previous identical sequences or patterns. Thus, the market reacts identically each time it encounters the same psychology as revealed by the price sequence. Many academic observers of charting are skeptical of the validity and effectiveness of most of the charting systems. However, as will be discussed later in the section regarding weak form market efficiency, we will discuss some exceptions in the academic literature. Increasing numbers of studies have been supportive of certain technical tools in the more recent literature. In any case, some of the better known charting systems or theories are reviewed below.

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The Dow Theory The Dow Theory, first suggested by Charles Dow in the late 1800's, examines

trends in the market and classifies them into three basic types: 1. Primary Trends: Commonly called bear or bull markets, these trends

represent the main component of the Dow Theory. They are long term, typically around 4 years in duration.

2. Secondary Movements: Referred to as corrections, these movements last only a few months. Typically, a secondary downward movement in a bull market is referred to as a correction and a secondary upward movement in a bear market is referred to as a bear trap.

3. Tertiary Moves: Daily fluctuations that are considered random variations. Although these movements are not essential to the theory, they are plotted to delineate primary and secondary trends.

A line chart may be plotted using each day's closing (or opening, or high, or low)

price, and connecting them with a line. Short-term trends may only represent a secondary movement, however a long-term trend, with each secondary trend failing to reach a new bottom may be considered a bull market.

A key aspect to the Dow Theory is the combination of both the DJ Industrials

Average and the DJ Transportations Index. The theory holds that the DHIA indicator is more meaningful when it is consistent with the Transportation Index. Presumably, the DJIA indicates productivity while the Transportation Index indicates demand reflected by goods in the order and transportation process. The more recent Elliott Wave Principle is based on five "waves" which are analogous to movements in the Dow theory.

The Elliott Wave

The Elliott Wave Theory, developed by a retired railroad engineer by the name of Ralph Nelson Elliott in the 1930's, was popularized by Robert Prechter in the early 1980's. It seemed that Prechter had used this obscure theory to correctly predict the bull market of the early and mid 1980's, and more interestingly, predicted the October 1987 crash two weeks before its occurrence. The theory lost credibility when, after the crash, Prechter predicted that the market would crash again, reducing the Dow to around 400. Nonetheless, the theory still maintains a following.

The Elliott Wave Theory holds that the market moves in cycles composed of five

waves. Three of the five waves are termed impulse waves and indicate the overall trend of the market while the other two are termed corrective waves. One is able to determine the future direction of the market by determining which are the impulse waves and which are the corrective waves.

Bar Charts

Bar Charts are frequently used to analyze individual securities. Vertical bars represent each day's price range, from highest to lowest. Price bar charts are frequently accompanied by bar graphs along the bottom indicating volume of shares traded. A

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chartist will plot prices over time for a security to locate relevant or recurring patterns. Such patterns may include the "head and shoulders", "triangle" or the "flag" figures. Practitioners and observers of charting generally regard the practice as an art rather than as a science.

Figure 1 provides an example of a bar chart taken from 10-minute intervals over

the period May 24-25, 2007. This chart depicts high and low values realized for the S&P 500, with left and right notches indicating open and close values.

Figure 1: Sample Bar Chart from Barchart.com

Moving Averages

Moving averages are frequently used as a reference point to gauge or smooth daily fluctuations. Daily prices are compared to a moving average of a specified number of historical prices. For example, one very simple rule holds that if current prices rise above a falling moving average, they are expected to drop back towards the moving average; selling is suggested. Buying is suggested when the moving average flattens out and the stock's price rises above the moving average. There are numerous other moving average rules.

Moving averages may be computed for any number of price data points. For

example, consider the following sequence of daily closing prices for a given stock over a period of time:

12 14 17 13 14 19 22 17 11 18 16 22 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9 t=10 t=11 t=12

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The following represents the sequence of three-day moving averages for the above price sequences: NA NA 14.3 14.7 14.7 15.3 18.3 19.3 16.7 15.3 15.0 18.7 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9 t=10 t=11 t=12 The simplest moving rule discussed above would suggest that days 4, 5, 8 and 9 are buying days while days 3, 6, 7, 10, 11 and 12 are selling days. Technical analysts make use of simple moving averages, exponential moving averages that weight more recent data more heavily. In addition, many of the moving average based rules are somewhat more complicated than the simple one described above.

Theories Based on Behavior of Certain Groups

Other technical analytical systems focus on the behavior of other groups of market participants. For example, Contrary Opinion Theories propose doing the opposite of what some particular group of investors is doing. One Contrary Opinion Theory is the Odd Lot Theory, which assumes small investors are usually wrong; thus, one should follow a strategy contrary to small investors. Odd lot transactions (less than 100 shares) require higher brokerage commissions and are usually placed by individual investors with limited funds. Years ago, odd lot volume data was collected by odd lot traders on the floor of the NYSE, compiled and published in financial sections of many newspapers. A second contrary opinion theory is based on short-sales volume. This Short-sales Contrary Opinion theory assumes that a high level of outstanding short sales is a sign of increased future demand in order to cover outstanding short positions. The theory asserts that this future increase in demand will bid up the prices.

Breadth-of Market Statistics are used to study the underlying strength of market

advances or declines. For example, before the October 1987 crash, the 30 DJIA blue-chip stocks were rising, but the majority of lesser-known stocks were declining. One easy way to study the breadth of the market is to compare the number of issues in a large market or index, such as the NYSE or Wilshire 5000 that advanced in price to the number that declined in price. Subtracting the issues whose prices declined from the number whose price advanced gives the daily net advances or declines. These daily net advances or declines are cumulated to obtain the breadth of market statistic. The breadth statistic may be negative during a bear market, or positive during a bull market. The direction of the breadth statistic, not the level is what is relevant.

All of these technical analysis techniques attempt to measure the supply and

demand of a security, and use that information to choose securities to buy or sell. These theories assume that shifts occur slowly over time, not instantaneously and can be charted and detected. Most technical analysis techniques are contrary to efficient market theories, which assert that new information is reflected in security prices instantaneously. Therefore, many financial academics believe technical stock analysis is not likely to generate higher than normal returns for investors. However, many individuals have become quite wealthy using technical tools. In addition, some observers of technical analysis feel that it is a useful timing devise when used with other fundamental and portfolio analysis tools to determine which securities to purchase.

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Other technical trading rules include the hemline theory (changes in stock prices

seem correlated with changes in dress hemlines), January rules (1. The market is stronger in January than in other months and 2. If returns are high in January, returns for the rest of the year will be high; if January returns are low, returns in subsequent months will be low). The market seems to increase during years when NFC teams win the Super Bowl and decline when the AFC wins (based on 22 correct calls from the first 25 Super Bowls). Presidents whose parties win election to the presidency (e.g., reelection) seem to indicate strong markets in the subsequent year. In addition, some investors pay particular attention to favorite "bellwether" stocks that they feel lead the market. Short interest may indicate either upward or downward potential, depending on whether an investor believes short sales indicate future demand for shares.

In many respects, technical analysis is like astrology or alchemy. Its validity is

impossible to disprove, yet many of its principles are difficult to justify. However, as was the case for astrology and alchemy earlier, technical analysis is frequently used and investors and has provided much insight into important phenomena. Understanding how traders and investors behave is obviously important to understanding and profiting from stock market behavior, though the extent to which the actual technical theories themselves benefit traders is debatable.

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B. Weak Form Efficiency Weak form efficiency tests are concerned with whether an investor might consistently earn higher than normal returns based on knowledge of historical price sequences. One can never prove weak form efficiency because there are an infinite number of ways to forecast future returns from past returns. However, one might argue that certain tests imply efficiency (or inefficiency) with regard to a certain sequence or pattern of prices. An investor armed with the knowledge of a test indicating a market inefficiency might expect to earn a higher than normal return, or face a market impediment preventing him from doing so. On the other hand, any test of market inefficiency is actually a joint test for that particular inefficiency and of some model describing normal returns in an efficient market. Thus, market efficiency cannot be fully rejected unless one is certain that the correct model for normal returns has been selected for that market. Furthermore, Grossman and Stiglitz [1980] argue that the existence of costly information retrieval and processing must lead to abnormal return sequences. Thus, markets cannot be fully efficient, and, perhaps, the appropriate benchmark for market efficiency tests should probably not be the hypothetical perfectly efficient market. In any case, some of the more significant weak form efficiency tests are described in the following paragraphs. Price Sequences and Relative Strength In one of the earlier weak form efficiency tests found a very weak relationship between historical and current prices - .057% of a given day's variation in the log of the price relative is explained by the prior day's change in the log of the price relative. On the other hand, another study by Fama and MacBeth [1973], after adjusting for risk, found no correlation in daily CAPM residuals. A typical testing format is as follows:

ttt rbar ++= 1 where tr represents the log of 1 plus a stocks return on a given day t, 1 tr the log of 1 plus the return on the day immediately prior, a and b regression coefficients and t the daily error terms in the regression. The sign of the b coefficient here is key. A negative value for b suggests that sequential returns or price changes are inversely correlated, indicating that stock prices are more likely to drift towards mean values than drift away. Positive values for b suggest that directions sequential price changes are likely to be sustained, indicating that price changes might carry momentum. However, one must remember that correlation coefficients are unduly influenced by extreme observations. This means that returns and logs of one plus returns are not normally distributed as required by OLS regression assumptions. One way to deal with this is to construct a runs test. For example consider the following daily price sequence: 50, 51, 52, 53, 52, 50, 45, 49, 54 and 53. The price changes might be represented by the following: (+++---++-), indicating four price runs. That is, there were four series of positive or negative price change runs. The expected number of runs in a runs tests if price changes are random is (MAX + MIN)/2, where MAX is the largest number of possible runs (equals the number of prices in the series and MIN is the minimum number (1). The number of runs consistent with random sequences in our example is 10 =

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(9+1)/2. More runs than this number suggests price reversals (mean reversion) and a smaller number of runs suggests positive correlations between sequential price changes (momentum). Thus, our example might be consistent with slight evidence of momentum. The actual levels of returns are unimportant; only the signs of returns are important, so that extreme observations will not unduly bias tests. In one test of daily price changes, Fama [1965] expected 760 runs based on the assumption that price changes were randomly generated, but only found 735 runs. This indicated a small positive correlation in daily stock price changes. Granger and Morgenstern [1972] found that high transactions costs seem to be related to runs - investors are unable to exploit a series because of brokerage commissions. There seem to be 2 to 3 times as many reversals of price trends as continuations based on transaction-to-transaction price data. This would be consistent with mean reversion if not for a particular type of market dynamic. This inverse correlation might be because of unexecuted limit orders - for them to be executed the price has to reverse itself. All of the limit sell order has to be used up for the price to increase so a purchase is likely to be followed by a downtick (-) or no change at all (0). Relative strength tests are concerned with the current price of a stock relative to a historical price or the prices of other securities or indices. For example, one relative strength rule compares the current price of a stock to its average price over a given period of time. Levy [1967] found the following rule to be profitable: rank stocks based on the ratio (pt/pavg) where pt is the current price and pavg is the average price over the prior 27 weeks, buy the stocks in the highest 5th percentile and sell the stocks in the lowest 70th percentile. However, this rule may simply mean we buy the riskiest securities (high E[R]) and sell the lowest risk securities. One very early study found that relative strength rules worked no better than random purchases of stocks after adjusting for transaction costs and even worse given risk adjustments. They suggest that the success of the filter rule discovered by Levy is due merely to selection bias and the fact that Levy tested 68 different trading rules. That is, one of the 68 trading rules must surely work on his limited size sample of prices. Filter Rules and Market Over-reaction A filter rule states that a transaction for a security should occur when its price has changed by a given percentage over a specified period of time. For example, Alexander [1961] suggested that one might infer that a price increase of a given proportion indicated an upward trend, and should serve as a purchase signal. He concluded that there were identifiable trends in security prices, though his study was somewhat unsophisticated by today's standards. Two other early studies found that only a few filter rules worked, but only slightly, and were not able to cover transactions costs. Profitability of these rules seem to be related to daily correlations. The correlation and filter rules seemed to work slightly better in Norway and Sweden, where stronger correlations seem to exist. These markets are less liquid and transactions costs are significantly higher in Norwegian markets than in American markets. The rules did not seem to work well enough to cover the particularly high transactions costs.

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Market overreactions to information and abnormal returns can be realized by buying losers and selling winners. One study indicated that buying stocks that performed poorly in a prior 3-5 year period and selling those that performed well would have generated higher than normal returns in subsequent 3-5 year periods. Most significant mean reversion tendencies seem to hold for the month of January. Other studies found similar results for market indices in various countries. However, later studies have suggested that these results might be explained by systematic risk or correlated with the small firm or January effects (discussed later). Another study found that the level of short-term inertia displayed by stocks is inversely related to firm size. Thus, if smaller firms have smaller investor followings and more restricted information availability, this apparent inertia might suggest that the market is relatively slow to react to information. Fama and French [1988] find evidence of negative autocorrelations in three to five year stock returns while Poterba and Summers [1988] find evidence of long term mean reversion in security prices. Their results would suggest that investors expect that investors overact to new information regarding stocks, though in the longer term, trends reverse as investors realize that they have overreacted. However, other studies have concluded that these findings of abnormal returns are due largely to statistical methodology employed in these papers. Moving Averages and Resistance Levels

Brock, Lakonishok and LeBaron [1992] demonstrated evidence suggesting that certain moving average rules and rules based on resistance levels produced higher than normal returns when applied to daily data for the Dow Jones Industrial Average from 1897 to 1986. However, Sullivan, Timmerman and White (1997) tested their findings on updated data and found that the best technical trading rule does not provide superior performance when used to trade in the subsequent 10-year post-sample period. Calendar Effects Numerous studies have confirmed a "January Effect". This "effect" is that returns for the month of January tend quite consistently to exceed returns for any other of the eleven months. This January effect seems most pronounced during the first five trading days in January and the last trading day in December. However, the January effect continues to persist throughout the month. Additionally, this January effect seems to have a greater effect on the shares of smaller companies (which are frequently held by individual investors) than on the shares of larger firms (frequently held by institutional investors). It appears that much of the January effect can be explained by the tendency for December transactions to be seller initiated and execute at bid prices while January transactions are buyer initiated and execute at offer prices. However, the latter two papers argue that the January effect is large enough that it would exist even if all transactions executed at bid prices. None of these papers explains the size of the January effect. One explanation for this January effect is year-end tax selling - investors sell their "losers" at the end of the year to capture tax write-offs. This year-end tax selling forces down prices at the end of the year. They recover early in the following year, most

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significantly during the first five trading days in January (and the last trading day in December). One study found that stocks selling at their annual low at year's end out performed the market in January by 8%. However, this increase barely covered transaction costs for investors wishing to exploit this price increase (He assumed a high transaction cost). Nonetheless, an investor wishing to purchase securities in late December may benefit from this knowledge. This year-end selling hypothesis seems supported by abnormally high trading volume in December. Adding further credibility to the year-end tax selling hypothesis, other studies have also found that "losers" outperform "winners" in January of the subsequent year and that firm returns in January are directly related to losses in the prior year. Furthermore, there seem to be consistent January effects for low-grade corporate bonds and in shares of companies that issue these bonds. This effect does not seem to hold for high-grade corporate bonds or for the shares of the companies that issue these bonds. This seems to further support the tax loss selling hypothesis for the January effect. Contrasting the tax explanations for the "January Effect" are studies demonstrating that this effect exists in markets whose tax years differ from the calendar year. Furthermore, the January effect in Canada existed before the introduction of a capital gains tax. However, one might argue that U.S. markets are sufficiently influential in world markets that year-end tax selling in the U.S. may drive prices in other markets. On the other hand, studies have demonstrated a January effect in U.S. markets during the period 1877-1916, prior to the introduction of U.S. income taxes. Furthermore, Kihn [1996] presents evidence that municipal bond issues, which are free from federal taxation, experience a significant January effect. Thus, despite the popularity of the tax explanation, there still remain serious doubts regarding its ability to explain the January effect. Some market observers have also suggested that funds will "window dress" at year-end by buying winners (stocks that performed well earlier in the year) and by selling losers. These transactions occur at the end of the year so that their clientele can see from year-end financial statements that their funds held high-performing stocks and did not hold losers. The buy pressure on winners and sell pressure on losers might be expected to cause losers to underperform winners during the period of window dressing; in January after the period of window dressing losers will outperform winners. However, most institutions will report their holdings to their clients more than once per year and many will report based on fiscal years other than from January 1 to December 31. Furthermore, winners still realize higher January returns than in any other month; just not as high as losers. If the "window dressing" hypothesis explains the January effect better than the tax-selling hypothesis, one should expect that shares held by institutions should outperform shares held by individuals during the month of January. First, we note that the January effect is more pronounced for smaller firms than for larger firms (smaller firms are more likely to be held by individual investors). Second, another study found that the January effect is far more pronounced for companies with significant numbers of individual shareholders than those companies with significant numbers of institutional

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investors. Thus, the tax-selling hypothesis seems to explain the January effect better than the "window-dressing" hypothesis. However, since there are still many holes in the tax-selling hypothesis, it appears that the January effect remains an anomaly. Two studies suggest that the January anomaly is more applicable to low priced shares than to higher priced shares. They argue that lower priced shares have higher proportional transactions costs and experience infrequent trading. Bhardwaj and Brooks find that the January effect could not have been exploited by typical investors during the period 1977-86 due to wide bid-ask spreads and high transactions costs. Thus, they argue that at least part of the January effect is related to transactions costs. While these studies suggest that investors may not be able to exploit this January effect, they do fully explain why the January effect exists. Several research papers have documented a "day of the week" effect. These papers have all suggested that end of week returns exceed Monday returns. Although this effect seems much less significant than the January effect, it still has not been adequately explained. Part of the effect might be attributed to the delay between stock trading and check settlements. In addition, part of the effect might be due to increased individual investor sell activity on Mondays. The Small Firm and P/E Effects There also exists substantial evidence that smaller firms outperform larger firms. For example, if one were to rank all NYSE, AMEX and NASDAQ firms by size, one is likely to find that those firms that are ranked as smaller will outperform those that are ranked larger. This effect holds after adjusting for risk as measured by beta. However, other measures of risk may be more appropriate for smaller firms that may not have well-established trading records due to market thinness or due to "newness" when compared to typical larger firms. Furthermore, transactions costs for many smaller firms may exceed those for larger firms, particularly when they are thinly traded. There is also evidence that these abnormally high returns for smaller firms are most pronounced in January. Although Fama and French [1992] find a significant size effect in their study of the CAPM over a fifty-year period, they do not find a size effect during the period between 1981 and 1990. This may suggest that the size effect either no longer exists or was merely a statistical artifact prior to 1981. Two studies found that abnormal returns in January are more related to share price effects than to firm size; in fact, after controlling for share prices (and noting the higher proportional transactions costs on lower priced shares), the size effect disappears as a partial explanation of the January effect. Basu [1977] and Fama and French [1992] find that firms with low price to earnings ratios outperform firms with higher P/E ratios. Fama and French find that the P/E ratio, combined with firm size predict security returns significantly better than the Capital Asset Pricing Model. The IPO Anomaly As we discussed earlier, The IPO anomaly refers to three unusual pricing patterns associated with Initial Public offerings of equities:

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1. Short-term IPO returns are abnormally high. 2. IPOs seem to underperform the market in the long run. 3. IPO underperformance seems to be cyclical. Sports Betting Markets Sports betting markets potentially have much in common with stock markets. There is some evidence of persistent inefficiencies in sports betting markets. For example, favorites in horse races outperform long shots while the opposite is true for baseball betting. Several observable variables in addition to the spread can be used to improve the outcomes of professional basketball games. Several studies suggest that certain strategies can be used to improve professional football betting. Summary Generally, statistical studies seem to indicate that stock markets are efficient with respect to historical price sequences. However, one must realize that an infinite number of possible sequences can be identified with any series of prices. Clearly, many of these series must be associated with higher than normal future returns. Statistical studies have certainly found evidence of predictability in stock prices, normally referring to such findings as anomalies.1 However, when research finds a sequence that leads to higher than normal returns, one must question whether the abnormal return result is merely a statistical artifact due to data mining. William Schwert [2003] was quoted:

These [research] findings raise the possibility that anomalies are more apparent than real. The notoriety associated with the findings of unusual evidence tempts authors to further investigate puzzling anomalies and later try to explain them. But even if the anomalies existed in the sample period in which they were first identified, the activities of practitioners who implement strategies to take advantage of anomalous behavior can cause the anomalies to disappear (as research findings cause the market to become more efficient).

Richard Roll [1992], in a blunt comment, stated:

I have personally tried to invest money, my clients and my own, in every single anomaly and predictive result that academics have dreamed up. That includes the strategy of DeBondt and Thaler (that is, sell short individual stocks immediately after one-day increases of more than 5%), the reverse of DeBondt and Thaler which is Jegadeesh and Titman (buy individual stocks after they have decreased by 5%), etc. I have attempted to exploit the so-called year-end anomalies and a whole variety of strategies supposedly documented by academic research. And I have yet to make a nickel on any of these supposed market inefficiencies. Clearly, technical analysis has its share of critics. For example, Warren Buffet

was quoted saying I realized technical analysis didn't work when I turned the charts upside down and didn't get a different answer. In any case, most apparent incidences of 1 Most notably, the IPO effect, calendar effects, size effects and momentum/reversion effects.

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mispricing seem eliminated by transactions costs. The primary exceptions to weak form market efficiency seem to be IPO effect, probably the January effect, perhaps the small firm effect, and perhaps the P/E effect. There seems to be little agreement as to why these effects persist or even if the latter two do exist; they remain anomalies.

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C. Back-testing Momentum and Mean Reversion Strategies

Suppose an analyst was interested in whether there existed a relationship between the daily return on a security and its return on a prior day. For example, consider the following sequence of daily closing prices for a given stock Q, from which we compute returns: Table 1 Return Sequences

Date t Pricet Returnt Returnt-1 02/01/07 1 49 02/02/07 2 50 .020408 02/03/07 3 51 .020000 .020408 02/04/07 4 52 .019607 .020000 02/05/07 5 55 .057692 .019607 02/06/07 6 57 .036363 .057692 02/07/07 7 58 .017543 .036363 02/08/07 8 59 .017241 .017543 02/09/07 9 58 -.016949 .017241 02/10/07 10 55 -.051724 -.016949 02/11/07 11 53 -.036363 -.051724 02/12/07 12 52 -.018867 -.036363

The prices given above assume that the stock traded each day, including weekends. Although there are many ways to determine the nature of the relationship between the return on a security and its prior day return, we will examine whether there exists a linear relationship based on a simple ordinary least squares regression of the form: rt = a + brt-1. We report regression results as follows:

17717758.0020142. += trr r-square = .57 (-.25959) (3.259325) SSE = .00450311 n = 10 d.f. = 8

Note that t-statistics are given in parentheses. Computations to obtain the t-statistics given above are described in Table 2. Given our standard error estimates se(a) = .007760684 and se(b) = .05606069533, we find our t-statistics to be t(a) = -.25959 and t(b) to be 3.2593. Suppose our two tests were expressed formally as follows:

H0: a = 0; H0: b = 0 HA: a 0; HA: b 0

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Table 2 Significance of Return Correlations

t Returnt Returnt-1 E[Rt] ,i ,i2

3 .020000 .020408 .013735 .00626 .000039234 .019607 .020000 .013420 .00618 .000038275 .057692 .019607 .013118 .04457 .001986846 .036363 .057692 .042510 -.00614 .000037787 .017543 .036363 .026049 -.00850 .000072358 .017241 .017543 .011525 .00571 .000032679 -.016949 .017241 .011291 -.02824 .0007975510 -.051724 -.016949 -.015095 -.03662 .0013416511 -.036363 -.051724 -.041934 .00557 .0000310212 -.018867 -.036363 -.030079 .01121 .00012569 SSE = .00450311

23679.056069.01003.

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Since we will assume that a and b follow student t distributions, we may compare

our t-statistic to critical values found in the t-distribution table, given the appropriate level of confidence and degrees of freedom. We are performing two tail tests for significance. Suppose that we wish to test for significance at a level of .95 (.025 on each tail of the distribution). In this case, the significance of our test is " = .025 and the number of degrees of freedom k equals 10 - 2 = 8. Our critical value for the test is found:

t10-2 = t0.025 = 2.306. While we fail to reject our first null hypothesis that a = 0, we reject our second null hypothesis that b = 0. Thus, our test does provide evidence that the returns for this security are linearly related to returns on prior days. More specifically, since b>0, we infer that stock returns are positively serially correlated, indicating momentum. Next, as a separate test for momentum in prices, we consider a runs test. First, starting with trading day 2, we have a positive price change, and this 7-day run continues through day 8. Then, we have a 4-day run of negative price changes. Thus, over 11 days, we have 2 runs, where (11+1)/2 = 6 would be expected if price changes exhibited no

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momentum. Thus, even though our price data set is too small to infer statistical significance, our runs test does provide some evidence supporting momentum in stock price sequences.

Thus, both of our tests seem to support the existence of momentum. If we felt that we could generalize these results to stocks in general (note that our sample consisted of only one stock over a very limited time span), we may use this relationship as the basis for a trading rule. However, before implementing such a trading rule, we should ensure that our result cannot be explained by factors that investors price (such as very high risk or inflation). In some cases, corrections can be made by subtracting out various types of systematic risk premiums from returns and performing regressions on abnormal return components. Also, one should be particularly careful to ensure that apparent weak form inefficiencies are not due to actual market inefficiencies such as the inability to trade, high transactions costs or high taxes. Such inefficiencies can easily consume all of the apparent profits from a trading rule.

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References Alexander, S. (1961): "Price Movements in Speculative Markets: Trends or Random Walks," Industrial Management Review, 2, pp. 7-26. Basu, S. (1977), Investment Performance of Common Stocks in Relation to their Price-Earnings Ratios, Journal of Finance, 32, 663-682. Brock, William, Josef Lakonishok, and Blake LeBaron, Simple technical trading rules and the stochastic properties of stock returns, Journal of Finance, vol. 47, 5, 1992, 1731-764. Edwards, Robert D. and John Magee (1997): Technical Analysis of Stock Trends, 7th ed., Amacom Books. Fama, E. (1965): "The Behavior of Stock Market Prices," Journal of Business, 38, 34-105. Fama, E. and M. Blume (1966), Filter Rules and Stock Market Trading Profits, Journal of Business, 39, 226-241. Fama, E. and K. French (1988), Permanent and Temporary Components of Stock Prices, Journal of Political Economy, 96, 246-273. Fama, E. and K. French (1992), The Cross-Section of Expected Stock Returns, Journal of Finance, 47, 427-465. Fama, E. and J. MacBeth (1973), Risk, Return, and Equilibrium: Empirical Tests, Journal of Political Economy, 71, 607-636. Grossman, Sanford and Joseph Stiglitz (1980): "On the Impossibility of Informationally Efficient Markets," American Economic Review, vol. 70, pp. 393-408. Granger, C.W. (1968) Some Aspects of the Random Walk Model of Stock Prices, International Economic Review, 9, 253-257. Granger, C.W. and O. Morgenstern (1972) Predictability of Stock Market Prices, Boston, Heath. Kihn, John (1996): "The Financial Performance of Low-Grade Municipal Bond Funds," Financial Management, 25, 2, pp. 52-73. Levy, Robert (1967): "Relative Strength as a Criterion for Investment Selection," Journal of Finance, 22, 595-610.

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Osborne, M.F.M. (1962): "Periodic Structure in the Brownian Motion of Stock Prices," Operations Research, vol. 10, pp. 345-379. Poterba J. and Lawrence Summers (1988): "Mean Reversion in Stock Prices," Journal of Financial Economics. Richard Roll in Volatility in U.S. and Japanese Stock Markets, selections from the First Annual Symposium on Global Financial Markets, Journal of Applied Corporate Finance, Vol. 5, 1, Summer 1992. Schwert, G. William, Anomalies and Market Efficiency, in The Handbook of The Economics of Finance, Constantinides, Harris, and Stulz, eds. (Amsterdam: Elsevier, 2003). Sullivan, Ryan, Allan Timmerman, and Halbert White, Data-Snooping, Technical Trading Rule Performance, and the Bootstrap, Unpublished UCSD Working Paper, December 8, 1997.

18

EXERCISES

1. Consider the following stock price sequence: t Pt 1 50 2 51 3 52 4 58 5 56

Compute three-day moving averages for days 3, 4 and 5. If one assumes that the three-day moving average price represents the true value of the stock because of the elimination of noise, should the investor buy or sell shares on days 3, 4 and 5?

2. Suppose an analyst was interested in whether there existed a momentum or mean reversion in month-to-month stock price series. The following table lists monthly prices for Stock X over a one-year period.

Date t Pricet 01/31/06 1 89 02/28/06 2 100 03/31/06 3 91 04/30/06 4 112 05/31/06 5 95 06/30/06 6 97 07/31/06 7 108 08/31/06 8 97 09/30/06 9 98 10/31/06 10 95 11/30/06 11 103 12/31/06 12 102

a. Calculate returns for each month. b. Calculate the variance of monthly returns. c. Calculate the covariance between returns for each month and the prior month. d. Calculate a regression coefficient (b) to determine evidence for mean reversion or

momentum. e. Does the stock price exhibit evidence for mean reversion or momentum? f. Is the b coefficient in part d statistically significant? g. If your answer for part f is yes, does a one-month decline in price suggest that

investors should buy or sell the stock at the beginning of the subsequent month? h. Comment on the reliability of your back-test. i. Is there evidence of mean reversion or momentum based on a runs test? j. Calculate a 5-month moving average price sequence for this series.

19

Solutions 1. Three-day moving averages are computed as follows:

t Pt MAt 1 50 NA 2 51 NA 3 52 51 4 58 53.67 5 56 55.33

Our three-day moving average would suggest that investors sell the stock on each of days 3, 4 and 5. 2.a. Returns for each month are calculated as follows:

t Pricet Returnt 1 89 2 100 0.12359551 3 91 -0.09 4 112 0.23076923 5 95 -0.15178571 6 97 0.02105263 7 108 0.11340206 8 97 -0.10185185 9 98 0.01030928

10 95 -0.03061224 11 103 0.08421053 12 102 -0.00970874

b. Most of our calculations will involve periods t=3 to t=12 since we have complete data for these dates. First, we calculate the average of returns from t=3 to t=12. This average equals .020908942. The variance of monthly returns, based on a population estimate is .012711. See the Appendix to Chapter 4 for a listing of appropriate formulas.

c. The covariance between returns for each month and the prior month are calculated as follows:

t Returnt Returnt+1 1 2 0.123595506 3 -0.09 0.123595506 4 0.230769231 -0.09 5 -0.151785714 0.230769231 6 0.021052632 -0.151785714 7 0.113402062 0.021052632 8 -0.101851852 0.113402062 9 0.010309278 -0.101851852

10 -0.030612245 0.010309278 11 0.084210526 -0.030612245 12 -0.009708738 0.084210526

The covariance from t=3 to t=12 is computed to be -0.008562423. Again, see the appendix to Chapter 4 for an appropriate covariance formula.

20

d. The regression coefficient (b) is simply the covariance between returns divided by the variance of monthly returns = -0.008562423/.012711 = -0.673627379.

e. This negative beta is evidence for mean reversion in stock prices. f. We compute our t-statistic for our beta coefficient as follows:

t Returnt Returnt-1 E[Rt] ,i ,i2

3 -0.09 0.12359551 -0.06159396 -0.028406 0.0008069 4 0.23076923 -0.09 0.082289818 0.1484794 0.0220461 5 -0.15178571 0.23076923 -0.13378912 -0.0179966 0.0003239 6 0.02105263 -0.15178571 0.123910367 -0.1028577 0.0105797 7 0.11340206 0.02105263 0.007481725 0.1059203 0.0112191 8 -0.10185185 0.11340206 -0.05472738 -0.0471245 0.0022207 9 0.01030928 -0.10185185 0.09027355 -0.0799643 0.0063943 10 -0.03061224 0.01030928 0.014718742 -0.045331 0.0020549 11 0.08421053 -0.03061224 0.0422846 0.0419259 0.0017578 12 -0.00970874 0.08421053 -0.03506316 0.0253544 0.0006428 SSE= 0.0580463

2389207.12711.

8/0580463.)(

2)( 2 ===

tt rrnSSE

bse

t = -.673627379/.2389207 = -2.8194606 < -2.306 Since the t-statistic is negative and less than its critical value at the .025 level, the

evidence that mean reversion in stock returns is statistically significant at the .025 (95%) level. Thus, the answer is yes. g. Mean reversion in stock returns suggests that the investor should purchase

immediately after a decline in prices. A one-month decline in price implies an expected increase in the subsequent month. Thus, the decrease implies that the investor should purchase the stock at the beginning of the subsequent month.

h. Perhaps the single most significant drawback to this particular test is its small and unrepresentative sample set. First, only one stock was used. Second, the test was based on only 10 data points, all from a single year.

i. We can work with 11 price changes, from t=2 to t=12. The price changes are + - + - + + - + - +. Thus, there are 9 runs out of a possible 10. The expected number of runs in a random series of 10 possible is (10+1)/2 = 5.5. Since the actual number of runs exceeds this expected value, this series reflects evidence of mean reversion.

j. Beginning with the 5th month, moving average prices are 97.4, 99, 100.6, 101.8, 99, 99, 100.2 and 99.

21

Appendix A: A Brief Review Of Hypothesis Testing In this section, we discuss the process of induction to form testable hypotheses or

theories from specific observations. These hypotheses or theories are useful if they provide a means to make meaningful predictions. Normally, testing a theory involves the collection of additional observations to determine whether they support the theory's predictions. If the additional observations do not confirm the predictions, then one has grounds for rejecting the theory. The observations collected to test a theory are usually represented by numbers or data. In most cases, statistical inference concerns the generalization of sample results to a population.

In many instances, one may make use of statistical inference to test a hypothesis. By convention, a hypotheses test usually involves formulation of a null (or maintained) hypothesis along with a competing alternative (research or challenging) hypothesis. The null hypothesis H0 usually is the claim that the population parameter equals some maintained value (note that null frequently implies no difference, no impact or nothing). The null hypothesis normally includes an equality sign or either or signs. The alternative hypothesis HA is the claim that the population parameter differs from the maintained value. The alternative hypothesis normally includes a strict inequality sign. Such tests are usually structured in a conservative manner such that the burden of proof is on the alternative hypothesis. One supports the research or alternative hypothesis by demonstrating the null hypothesis to be false (rejecting the null hypothesis). One rejects the null hypothesis only when the probability of its being true is sufficiently low (the conventional probability, known as a level of significance, is .05 or .01). In some instances, the appropriate level of significance for a hypothesis test can be based on the relative costs of rejecting the null hypothesis when it is true or accepting it when it is false.

One might list the steps of a typical statistical hypothesis test as follows: 1. Define the null hypothesis, H0. 2. Define the alternative hypothesis, HA. 3. Determine a level of significance, , for the test. 4. Determine the decision rule or test statistic along with acceptance or rejection

regions or critical value based on . 5. Perform computations. 6. Form conclusions.

The decision rule or test statistic is a given function of a measurement drawn from the sample on which the statistical decision will be based. The rejection region consists of those values of the test statistic that will lead to rejection of the null hypothesis. The critical value marks the boundary between the acceptance and rejection regions.

An experiment involving a given sample drawn from a population has some probability of resulting in an erroneous conclusion. Thus, one's hypothesis test may lead to an incorrect acceptance of the null hypothesis (Type I error) or an incorrect rejection

22

of the null hypothesis (Type II error). The power of a test refers to the probability of not committing a Type II error. This is equivalent to the probability of accepting the alternative hypothesis when it is correct. A test is considered to be superior when its power is higher.

Statistics are most useful for empirical studies in finance. This chapter provides, at best, a very superficial overview of a few of the applications of statistical methodology in finance. The reader is advised to consult a more comprehensive text (such as those listed in Suggested Readings at the end of this chapter) for a more detailed presentation of statistical methodology and its applications to financial problems. Hypothesis Testing: Two Populations

Here, we are concerned with comparing two means, 1 and 2 for populations 1 and 2 with standard deviations 1 and 2. We shall assume that our samples are independent and drawn from populations whose data are normally distributed. Our test will be based on samples of sizes n1 and n2. The samples will have means and variances equal to x1 and x2 and s21 and s22, respectively. We will base our testing methodologies on test statistics and distributions somewhat different from those used earlier. Suppose that we wanted to test whether the means of two populations were different based on samples drawn from those populations. Our hypotheses and test statistics might be as follows:

H0: 1 = 2 HA: 1 2

(A.9)

+

++

=

21

21

21

222

211

2121

)2()1()1(

)()(

nnnn

nnsnsn

xxt

where s21 and s22 are the sample variances. If we are testing whetherx1 andx2 are equal, then our hypothesized difference in means 1 - 2 = 0 is used for computing our test statistic. Our test statistic assumes that our data follows a student-t distribution.

A variety of other types of tests involving samples from two populations can be constructed as well. For example, tests can be developed to determine whether variances differ, other tests can be based on samples with matched pairs of observations, and so on. A statistics or econometrics text can be consulted to provide additional testing methodologies. Introduction To The Simple OLS Regression

Regressions are used to determine relationships between a dependent variable and one or more independent variables. A simple regression is concerned with the relationship between a dependent variable and a single independent variable; a multiple regression is concerned with the relationship between a dependent variable and a series of independent variables. A linear regression is used to describe the relationship between the dependent and independent variable(s) to a linear function or line (or hyperplane in the

23

case of a multiple regression).

The simple Ordinary Least Squares regression (simple OLS) takes the following form: (A.10) yt = b0 + b1xt +i,t

The ordinary least squares regression coefficients b0 and b1 are derived by minimizing the variance of errors in fitting the curve (or m dimensional surface for multiple regressions involving m variables). Since the expected value of error terms equals zero, this derivation is identical to minimizing error terms squared. Regression coefficient b1 is simply the covariance between y and x divided by the variance of x; b1 and b0 are found as follows:

(A.11)

== n

ii

n

iii

x

yx

xx

yyxxb

1

2

12

2,

1

)](

))(()(

(A.12) xbyb 10 = Appropriate use of the OLS requires the following assumptions: 1. Dependent variable values are distributed independently of one another. 2. The variance of x is approximately the same over all ranges for x. 3. The variance of error term values is approximately the same over all ranges of x. 4. The expected value of each disturbance or error term equals zero.

Violations in these assumptions will weaken the validity of the results obtained from the regression and may necessitate either modifications to the OLS regression or different statistical testing techniques.

There are numerous types of regressions depicting different types of relationships among variables. Table A.1 provides details on some of these different types of regressions. A simple regression is concerned with the relationship between a dependent variable and a single independent variable. Regression coefficients b0 and b1 represent the vertical intercept and the slope in the statistical linear relationship between the dependent variable yi and the independent variable xi. Thus the vertical intercept b0 represents the regression's forecasted value for yi when xi equals zero and the slope of the regression b1 represents the change in i (the value forecast by the regression for yi) induced by a change in xi. The error term i represents the vertical distance between the value i forecasted by the regression based on its true value yi; that is, i = yi i. The OLS regression minimizes the sum or average of these error terms squared. The size of the sum of the squared errors (often called SSE or, when divided by (n-2), the variance of errors 2 ) will be used to measure the predictive strength of the regression equation. A

24

regression with smaller error terms or smaller 2 is likely to be a better predictor, all else held constant.

Table A.1 Classes of Ordinary Least Squares Regressions

By

Number of

Variables

By Shape

of Curve

Example

Simple Linear iii xbbY ++= 10

Multiple Linear iiii zbxbbY +++= 210

Simple Curvi-linear*

i

bii xaY +=

Simple Log-linear*

iii xbbY ++= )log()log( 10

Multiple Curvi-linear*

i

bi

bii zaxaY ++=

)2(2

)1(1

Multiple Log-linear*

iiii zbxbaY +++= )log()log()log( 211

Simple Non-linear**

i

bii xaaY ++= 10

Multiple Non-linear**

i

bi

bii zaxaaY +++=

)2(2

)1(10

* Curvi-linear regressions may be transformed into linear regressions. In these examples, the transformation is completed by finding the log of both sides, while ignoring the error term, since its expected value is zero. ** Non-linear regressions cannot be transformed into linear regressions.

Once we have determined the statistical relationship between yi and xi based on our OLS, our next problem is to measure the strength of the relationship, or its significance. One of the more useful indicators of the strength of the regression is the coefficient of determination or 2 statistic. The coefficient of determination (often referred to as r-square) represents the proportion of variation of variable y that is explained by its regression on x. It is determined as follows:

25

(A.13)

== n

i

n

iii

n

iii

yx

yxyx

yyxx

yyxx

1 1

22

1

2

22

2,2

,

)]([)]([

)])(([)(

This coefficient of determination may also be expressed as either of the following:

(A.14) yin Variation Total

Regression by the Explainedy in Variation Total2, =yx

(A.15)

=

= n

ii

n

i

n

iii

yx

yy

yy

1

2

1 1

22

2,

)(

)(

The sum (yi -y )2 represents total variation in y; the sum 2 represents the variation in y not explained by the regression on x.

Assume that there exists for a population a true OLS regression equation yi = 0 + 1xi + i representing the relationship between yi and xi, without measurement or sampling error. However, we propose the regression yi = b0 + b1xi + i, whose ability to represent the true relationship between yi and xi is a function of our ability to measure and sample properly. Our sampling coefficients b0 and b1 are merely estimates for the true coefficients 0 and 1 and they may vary from sample to sample. It is useful to know the significance of each of these sampling coefficients in explaining the relationship between yi and xi.

Our estimate b1 for the slope coefficient 1 may vary from regression to regression, depending on how our sample varies. Our estimates for b1 will follow a t-distribution if our sample of yi's is large or normally distributed; if our sample is sufficiently large, our estimates for b1 may be characterized as normally distributed. One potential test of the significance of our coefficient estimate b1 is structured as follows:

H0: 1 0 HA: 1 > 0

Our null hypothesis is that y is unrelated or inversely related to x; our alternative

hypothesis is that y is directly related to x. The first step in our test is to compute the standard error se(b1) of our estimate for b1 as follows:

26

(A.16)

=

=

== n

ii

n

ii

x

z

xx

n

z

nbse

1

2

1

2

2

2

1

)(

21)(

The standard error for b1 is, in a sense, an indicator of our level of uncertainty

regarding our estimate for b1. The numerator within the radical indicates the variability unexplained by the regression; the denominator indicates total variability. Our next step is to find the test statistic for b1. This is analogous to standardizing or finding the normal deviate in our earlier hypothesis tests:

(A.17) )(

)(1

11 bse

bbt =

We next compare this test statistic to a critical value from a table representing the

t-distribution (See Table A.4) or representing the z-distribution (See the z-table in A.3).

The process for determining the statistical significance of the vertical intercept b0 is quite similar to that for determining the statistical significance for b1. We first designate appropriate hypotheses, such as those that follow:

H0: = 0 HA: 0

The primary difference in the process is in determining se(b0):

(A.18)

= n

ii

n

ii

n

ii

xxn

xn

z

bse

1

2

1

21

2

0

)(

2)(

Next, we find our t-statistic as follows:

(A.19) )(

)(0

00 bse

bbt =

We then compare our t-statistic to the appropriate critical value just as we did

when testing the significance of the slope coefficient. This particular test involves two tails, since our alternative hypothesis is a strict inequality. Be certain to make appropriate adjustments to the critical value (for example, divide by 2 for two tailed tests) when making comparisons.

27

Table A.3: The z-Table z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 .0000 .0040 .0080 .0120 .0159 .0199 .0239 .0279 .0319 .0358 0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753 0.2 .0793 .0832 .0871 .0909 .0948 .0987 .1026 .1064 .1103 .1141 0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517 0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879 0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224 0.6 .2257 .2291 .2324 .2356 .2389 .2421 .2454 .2486 .2517 .2549 0.7 .2580 .2611 .2642 .2673 .2703 .2734 .2764 .2793 .2823 .2852 0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133 0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389 1.0 .3413 .3437 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621 1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830 1.2 .3849 .3869 .3888 .3906 .3925 .3943 .3962 .3980 .3997 .4015 1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177 1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319 1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441 1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545 1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633 1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706 1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767 2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817 2.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857 2.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890 2.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916 2.4 .4918 .492 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4936 2.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4952 2.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964 2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974 2.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981 2.9 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .4986 3.0 .4986 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990

28

Table A.4: The t-Distribution Right-tail area, " df 0.100 0.050 0.025 0.010 0.005 1 3.078 6.314 12.706 31.821 63.657 2 1.886 2.920 4.303 6.695 9.925 3 1.638 2.353 3.182 4.541 5.841 4 1.533 2.132 2.776 3.747 4.604 5 1.476 2.015 2.571 3.365 4.032 6 1.440 1.943 2.447 3.143 3.707 7 1.415 1.895 2.365 2.998 3.499 8 1.397 1.860 2.306 2.896 3.355 9 1.383 1.833 2.262 2.821 3.250 10 1.372 1.812 2.228 2.764 3.169 11 1.363 1.796 2.201 2.718 3.106 12 1.356 1.782 2.179 2.681 3.055 13 1.350 1.771 2.160 2.650 3.012 14 1.345 1.761 2.145 2.624 2.977 15 1.341 1.753 2.131 2.602 2.947 16 1.337 1.746 2.120 2.583 2.921 17 1.333 1.740 2.110 2.567 2.898 18 1.330 1.734 2.101 2.552 2.878 19 1.328 1.729 2.093 2.539 2.861 20 1.325 1.725 2.086 2.528 2.845 21 1.323 1.721 2.080 2.518 2.831 22 1.321 1.717 2.074 2.508 2.819 23 1.319 1.714 2.069 2.500 2.807 24 1.318 1.711 2.064 2.492 2.797 25 1.316 1.708 2.060 2.485 2.787 26 1.315 1.706 2.056 2.479 2.779 27 1.314 1.703 2.052 2.473 2.771 28 1.313 1.701 2.048 2.467 2.763 29 1.311 1.699 2.045 2.462 2.756 30 1.310 1.697 2.042 2.457 2.750 4 1.282 1.645 1.960 2.326 2.576

Examples: The t value for 4 degrees of freedom that bounds a right-tail area of 0.025 is 1.960.