Purpose of this page is to review the various biases and errors that distort empirical studies on equity returns and financial market anomalies. Usually the problems are suggested and subsequently examined in isolation. The consensus so far appears to be that none of these distortions alone can account for the anomalies reported in the literature. The conclusion by many researchers seems to be that therefore the various biases and errors can be ignored for simplicity. Of course this is probably not how it works, because these problems may have a tendency to accumulate rather than cancel out.
Research should be a cumulative science. Lessons learned from past studies should be incorporated and become a new standard for future research. Reality shows that research tends to revert to the zero point of origin and starts all over again, with all the problems of the past and lessons not learned.
- database errors
- Obviously, database owners will attempt to ensure that databases are free of typing errors, etc. However, errors of various origins do exist even in todays electronic databases (one might argue that errors may increase, because we come to rely on the automated execution of formulas and programs, not spending enough time to evaluate the results). For example, Bhardwaj and Brooks, 1992 footnote 8 report that the famous CRSP financial database contains a number of errors with respect to failures to adjust for (reverse) stock splits and stock conversions following bankruptcy reorganizations. Of course, not adjusting prices for reverse stock splits results in a serious overestimation of returns for small price stocks that are most likely to experience reverse stock splits (for example, 1-4 equals +400%), and not adjusting for stock splits will cause large negative returns on particularly large price stocks most likely to experience stock splits (for example, 4-1 equals -75%).
- return measurement bias
- All returns are not created equal. One needs to consider very carefully the use of arithmetic average and geometric average returns, compounding assumptions, and the validity of return measures in a portfolio context.
- Arithmetic time averages of percentage returns will be higher for time series that are more volatile over time. In finance we immediately think of the basic principle: risk and return are related. Taking advantage of this phenomenon is an easy way to manipulate investment advertorials. Geometric average returns or time averages of log returns will allow a more fair comparison.
- Comparing returns from different sources is complicated by the possibility of different compounding assumptions. Only log returns (i.e. continuously compounded returns) can be compared easily.
- It is thus easy to understand that finance research today primarily uses log returns. However, log returns cannot be used in calculating average returns of portfolios of individual assets. The correct procedure here would consist of 3 steps: 1- calculated portfolio returns using weighted arithmetic averages of percentage returns; 2- calculate the time series investment performance of the portfolio using the compounded return formula (i.e. create a wealth index); 3- calculate the average log return over time as a measure of portfolio performance.
- errors-in-variables bias
- Adjustment of returns for risk is unavoidable in finance. Frequently, this is still based on the CAPM-beta approach. But CAPM-betas are not observed, merely estimated. Empirical studies show that estimates of beta are usually biased towards zero due to noise in short-term returns. Obviously, using these biased beta estimates will distort required risk adjustment.
- Attempts have been made to use statistical methods to correct for the errors-in-variables bias. Several studies have shown that using longer return-measurement horizons, for example annual returns (Kothari, Shanken and Sloan (1995)), significantly improves the reliability of beta estimates. In general, corrections for the EIV bias increase the beta explanatory power in cross-sectional return studies. However, the estimation results are sensitive to arbitrary choices in portfolio formation methods used in the studies.
- Marston and Harris (1993) show that the practice of using realized or ex post returns also distorts tests of the beta or market risk effect. Measures of expected or ex ante returns are strongly positively related to beta.
- infrequent trading bias, nonsynchronous trading bias, aggregation bias
- (For example: Fisher, 1966; Dimson, 1979; Cohen, 1978, 1979; Lo and MacKinlay, 1990; Stoll and Whaley, 1990; Mathuswamy and Whaley; 1994).
- Databases tend to report the most recent transaction prices. However, not all shares are traded continuously. Illiquid stocks may not trade for days, and the daily reported price is the last traded price from several days earlier. Observed daily returns for these stocks are set to zero, which of course is not necessarily a correct reflection of changes in (unobserved) underlying fundamental prices.
- Nonsynchronous trading will result in cross-correlation relationships between returns in liquid (traded) and illiquid (nontraded) stocks. Traded stocks will respond to new information very quickly, whereas nontraded stocks will reflect new information with a lag - once the next trade occurs. The result will be spurious lead-lag relations and autocorrelation in portfolio returns.
- bid-ask spread bias
- Blume and Stambaugh (1983) show that the arithmetic average of period returns using transaction prices is biased by random movements between (low) bid price and (high) ask prices relative to the underlying fundamental value. Using a simple model with A the ask price and B the bid price, the expected or average observed return is 1+E(Rt) = 0.25*[At/At-1]+0.25*[At/Bt-1] +0.25*[Bt/At-1] + 0.25[Bt/Bt-1]. The return on the underlying price is 1+Rt = [(At+Bt)/2]/[(At-1+Bt-1)/2]. Rewriting shows that the bid-ask spread bias equals E(Rt)-Rt = s2/(4-s2) with s = [A-B]/[(A+B)/2] the constant percentage or relative bid-ask spread. For a 10% bid-ask spread, the period return bias equals 0.3%.
- Bhardwaj and Brooks (1992) extend this model to take into account that in certain periods the observation of bid and/or ask prices is not random (i.e. probabilities not 0.5), for example in December (bid) and January (ask). They show the total bid-ask bias as s2*(1-p*q)/(4-s2) + (2*s*(p-q)/(4-s2), with p and q the excess probabilities (relative to 0.5) of bid transactions at time t-1 and time t.
- The bid-ask spread bias is closely related to the rebalancing problem, but not identical.
- Blume and Stambaugh (1983)
- Bhardwaj and Brooks (1992)
- rebalancing bias
- For example: http://www.efficientfrontier.com/ef/996/rebal.htm)
- Most empirical studies use fixed weights to calculate the average returns on portfolios for the selected measurement period (for example, monthly returns). Keeping weights fixed assumes that stocks with high and low returns in a given period are sold at the end of the period and the resources reinvested at the original weights. But during the observation period high (low) return stocks have actually increased (decreased) their weight in the portfolio. After rebalancing previously high return stocks are therefore consistently underweighted and previously low returns consistently overweighted. When period returns are mean reverting (due to random shocks, negative autocorrelation, non-synchronous trading, etc), this is actually a positive effect for the portfolio return because high (low) return stocks will on average have lower (higher) returns in the next period. The size of the rebalancing bias -- relative to buy-hold portfolio returns -- depends on the volatility of individual stock returns and the cross-correlation of returns.
- One could argue that rebalancing is a valid way of increasing returns. However, rebalancing incurs high(er) costs from frequent trading.
- Roll (1983)
- Conrad and Kaul (1993)
- Ball, Kothari and Wasley (1995)
- Barber and Lyon (1997)
- survivorship bias
- Survivorship bias arises when the database only includes stocks that have survived the historical period. Other, poor performing stocks or mutual funds that at one time existed are eliminated from the current dataset. As a result, observed investment returns are too high, because the poor and negative returns that would have been part of the historical investment experience have been eliminated.
- backlisting bias
- The major source of US accounting data for stock selection is COMPUSTAT. Kothari, Shanken and Sloan (1995) argue that COMPUSTAT adds firms and their historical data for several years, when the firms have good performance and satisfy certain selection criteria. Firms with poor performance are less likely to be added to the database. Furthermore, firms that fail to report statements due to financial distress, but recover, may retroactively report financial statements for the previous periods. These procedures tend to favor firms with high book-to-market values (low prices) and subsequent high returns (recovery).
- Kothari, Shanken and Sloan (1995)
- Breen and Korajczyk, 19
- Chan, Jegadeesh and Lakonishok (1995)
- Kim (1997)
- The results are mixed. However, these studies tend to collect missing data from alternative sources with a focus on larger firms where the selection bias is less important.
- new listing bias
- On average, new stock market listings tend to be small, young, high growth firms. Conversely, small stock portfolios may contain a selection bias towards these newly listed stocks. Any special phenomenon related to new listings (long-term underperformance, short-term outperformance) may affect the small stock portfolio, without reflecting the nature of small stock returns perse.
- Lyon, Barber and Tsai (1999)
- delisting bias
- Returns can only be calculated as long as price data is available. When the time series of price data ends, the usual practice is to assume that stocks are sold at the last reported price and the funds redistributed amongst the remaining stocks in a portfolio. The CRSP database seems to be missing data on many of the delisting prices against which stocks must be sold when unanticipated delistings occur. As a result the final return on particularly small and poor performing stocks (high book-to-market value) is too high. Average delisting returns on NYSE-AMEX and NASDAQ stocks are estimated at respectively -30% and -55% but reach a maximum at -100% (Shumway, 1997; Shumway and Warther, 1999).
- Shumway (1997)
- Shumway and Warther (1999)
- real time, look ahead bias
- Tests of market efficiency and trading strategies must be based on the information available to market participants in real time. Creating portfolios on the basis of normal December account period data is not possible in real time due to the information lag in the accounting reports (although, shorter today than historically). Simulation studies based on database December data and December portfolio assumption introduce a looking-ahead-bias, assuming information that is not yet known. Another consequence may be that empirical research concludes there is a lagged response of the market, where in fact no such lag exists.
- Equally important may be recognition of the fact that frequently firms are obliged to restate accounting data for previous periods due to errors and omissions. The result may be that empirical research concludes there are biased expectations or forecast incorporated in market behavior, where in fact the bias is introduced ex post.
- small sample bias
- Statistical tests of significance usually rely on the theoretical asymptotic properties of estimates and significance. In reality, economic research is usually confined to limited numbers of observations, not satisfying the requirement for asymptotic theory. Sometimes, researchers attempt to artificially increase the number of observations, for example by increasing the frequency of observation (not annual, but daily data; not independent, but overlapping observations). They run the risk of ignoring the crucial difference between "span" and "frequency".
- skewness bias
- Statistical tests of significance usually rely on the standard assumption of normal distributed variables. In fact, empirical results show that in many cases variables such as (abnormal) returns are not normal distributed but skewed (and also subject to heteroskedasticity). In these cases the normal statistical test of significance is biased, rejecting the null hypothesis too frequently.
- Lyon, Barber and Tsai (1999)
- in-sample, ex-post, data snooping analysis
- Many empirical studies suggest the existence of anomalies and predictable returns using historical analysis, assuming that investors in the past probably should have known..., and assuming that historical patterns will simply persist in the future. Alternative approaches based on (quasi-)real-time and out-of-sample analysis tend to show that predictability and systematic excess returns disappear. For example, Cooper, Gutierrez and Marcum (2005) examine whether a real-time investor could have used book-to-market, firm size, and momentum returns to generate profits. Whether profits are generated in this real-time out-of-sample analysis depends on the particular selection criterion chosen by the investor to choose from alternative strategies but even the most profitable real-time strategy time provides returns much lower than suggested in other studies. Nearly all other studies of real-time investment performances fail to show that the market is clearly beatable.
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