Robust Regressions 169
Martin and Simin provide a feeling for the magnitude of the absolute
difference between the OLS beta and the resistant beta using weekly returns
for 8,314 companies over the period January 1992 to December 1996. A
summary of the distribution follows:
Absolute Difference in Beta No. of Companies Percent
0.0 + to 0.3 5,043 60.7
0.3 + to 0.5 2,206 26.5
0.5 + to 1.0 800 9.6
Greater than 1.0+ 265 3.2
Studies by Fama and French find that market capitalization (size)
and book-to-market are important factors in explaining cross-sectional
returns.^7 These results are purely empirically based since there is no equi-
librium asset pricing model that would suggest either factor as being
related to expected return. The empirical evidence that size may be a
factor that earns a risk premia (popularly referred to as the “small-firm
effect” or “size effect”) was first reported by Banz.^8 Knez and Ready
reexamined the empirical evidence using robust regressions, more spe-
cifically the least-trimmed squares regression discussed earlier.^9 Their
results are twofold. First, they find that when 1% of the most extreme
observations are trimmed each month, the risk premia found by Fama
and French for the size factor disappears. Second, the inverse relation
between size and the risk premia reported by Banz and Fama and French
(i.e., the larger the capitalization, the smaller the risk premia) no longer
holds when the sample is trimmed. For example, the average monthly
risk premia estimated using OLS is –12 basis points. However, when 5%
of the sample is trimmed, the average monthly risk premia is estimated
to be +33 basis points; when 1% of the sample is trimmed, the estimated
average risk premia is +14 basis points.
(^7) Eugene F. Fama and Kenneth R. French, “The Cross-Section of Expected Stock
Returns,” Journal of Finance 47 (1992): 427–466, and Eugene F. Fama and Kenneth
R. French, “Common Risk Factors in the Returns on Stocks and Bonds,” Journal of
Financial Economics 33 (1993): 3–56.
(^8) Rolf W. Banz, “The Relationship between Return and Market Value of Common
Stocks,” Journal of Financial Economics 9 (1981): 3–18.
(^9) Peter J. Knez and Mark J. Ready, “On the Robustness of Size and Book-to-Market
in Cross-Sectional Regressions,” Journal of Finance 52 (1997): 1355–1382.