In their March 2011 paper entitled “The Shrinking Space for Anomalies”, George Jiang and Andrew Zhang investigate the robustness of ten well-known anomalies by iteratively “shrinking the stock space” in two ways to determine whether and how the anomalies really work. The ten anomaly variables are: size, book-to-market ratio, momentum, two liquidity measures, idiosyncratic volatility, accrual, capital expenditure, sales growth and net share issuance. The first way of “shrinking the stock space” involves: (1) ranking the universe of stocks by each of the ten anomaly variables into deciles; (2) iteratively trimming deciles from side of a variable distribution that a hedge portfolio would sell and the side that a hedge portfolio would buy; and, (3) retesting the strength of the anomaly associated with the variable after each iterative trimming. The second way of “shrinking the stock space” involves: (1) trimming from the sample stocks with the smallest market capitalizations and the most extreme book-to-market ratios until size, book-to-market and momentum no longer have significant four-factor alphas for value-weighting and equal equal-weighting (thereby “perfecting” the sample for the four-factor model); and, (2) retesting the strength of the anomalies associated with the other seven variables using the perfected sample. This approach obviates weaknesses in alpha measurement via the commonly applied but imperfect three-factor (market, size, book-to-market) and four-factor (plus momentum) risk models. Using firm characteristics and trading data for all non-financial NYSE, AMEX, and NASDAQ common stocks over the period July 1962 through December 2007, *they find that:*

- Hedge portfolios for the ten anomalies are long (short) stocks with extremely:
- Small (large) size, measured as market capitalization.
- Low (high) book-to-market ratios.
- High (low) return momentum, typically measured over the past year with a skip-month.
- Low (high) liquidity, typically measured as impact of trading or trading turnover.
- Low (high) idiosyncratic volatility.
- Low (high) accruals.
- Low (high) capital expenditures, measured as asset growth.
- High (low) sales growth.
- Low (high) net stock issuance, measured as secondary offerings minus buybacks.

- Regarding the importance of the portfolio weighting scheme used to exploit anomalies:
- Size, accruals and net stock issuance anomalies are statistically significant for both value and equal portfolio weightings.
- Momentum and idiosyncratic volatility anomalies are significant only for value portfolio weighting.
- Book-to-market ratio, liquidity, capital expenditure and sales growth anomalies are significant only for equal portfolio weighting.

- Regarding the importance of the long (undervalued) and short (overvalued) sides of anomaly variable distributions:
- Both long and short sides drive the book-to-market ratio and momentum anomalies.
- The long side drives the size and illiquidity anomalies.
- The short side drives the accrual, capital expenditure and sales growth anomalies.

- Regarding shortcomings of the four-factor risk adjustment model:
- Excluding the 6% stocks with the smallest market capitalizations “perfects” the four-factor model for value weighting, in that alphas for the value-weighted size, book-to-market ratio momentum hedge portfolios are no longer highly significant. After this exclusion, only the accrual, net stock issuance and idiosyncratic volatility anomalies remain robust.
- Excluding first the 8% stocks with the smallest market capitalizations, the 28% of stocks with the lowest book-to-market ratios and the 28% of stocks with the highest book-to-market ratios “perfects” the four-factor model for equal weighting, in that alphas for the equal-weighted size, book-to-market ratio momentum hedge portfolios are no longer highly significant. After these exclusions, only the accrual and net stock issuance anomalies remain robust.
- In other words (especially for equally weighted portfolios), the four-factor model reliably explains returns only for a subset of the U.S. stock universe. Within this subset, the sales growth and capital expenditures anomalies do not occur, suggesting that they are not independent of the size, book-to-market ratio and momentum anomalies.

In summary, *evidence indicates that successful exploitation of ten widely recognized stock return anomalies may depend on: (1) whether the selected strategy employs equal or value portfolio weighting; (2) whether the selected strategy focuses on overvalued or undervalued stocks per the anomaly definition, or incorporates both; and, (3) whether the strategy excludes parts of the available stock universe (such as the smallest stocks).*

Cautions regarding these findings include:

- The study uses gross, rather than net, returns. Including trading frictions for anomaly implementations could affect conclusions. Momentum (high portfolio turnover) and liquidity (focused on stocks that are expensive to trade) anomalies may be most affected.
- Testing of multiple anomalies on a single data set introduces data snooping bias, such that returns for the best ones incorporate luck. Collecting “documented” anomalies arguably concentrates this bias.