fama macbeth revisited

Journal of Economic and Social Measurement 32 (2007) 41–63 41IOS Press

Replication Section

Fama MacBeth 1973: Reproduction, extension,robustification

Heiko Bailera and R. Douglas MartinbaABN-AMRO, Delta One Trading London, 250 Bishopsgate, London, EC2M 4AA, UKTel.: +44 20 7678 5681, Fax: +44 20 7678 1659; E-mail: [email protected] of Statistics, University of Washington, Seattle, WA 98145, USAE-mail: [email protected]

The empirical three-step, cross-sectional regression method of Fama and MacBeth [9] (FM) usedclassical ordinary least squares regression, averages, and t-statistics to reach its conclusions. Unfortunately,averages and t-statistics, taken over different volatility regimes and fractions of outlying data, can beseverely biased. This paper replicates and extends FM’s results to recent time periods, analyzes the choiceof time period, and replaces the classical estimators with a theoretically well-justified robust estimator invarious parts of the three-step approach. While FM’s conclusions on non-linearity and non-beta relatedrisk could be confirmed, the conclusions of having, on average, a positive risk and return trade-off couldnot be confirmed.

1. Introduction

It is suggested that the reader have a copy of the original Fama-MacBeth [9] articleat hand while reading this paper.

The Fama-MacBeth [9] cross-sectional regression technique, referred to as FM inthis paper, is not only a historically important method, but also still one of the mostwidely used tools in basic empirical finance with main contributions in empiricallyvalidating the important capital asset pricing model (CAPM) of Sharpe [23] andLinter [15] or the ATP model of Ross [20]. The empirical FM approach is a veryintuitive three-step procedure and can be easily extended to time-varying factors(Elton and Gruber [6]; Campbell, Lo and MacKinlay [2]; Cochrane [5]). The three-step procedure works as follows:

S I: GivenN equities useordinary least squares (OLS) regression to fit thesingle-factor market model

Rt = α+Rmt · β + εt, t = 1, . . . ,T (1)

with Rt the historical return of the equity in excess of the risk-free ratecomputed over time periodst-1 to t andRmt, the corresponding historicalexcess returns of a market proxy. For each equityi = 1, . . . , N this resultsin an estimate of the CAPM beta-measure of market riskβi.

0747-9662/07/$17.00 2007 – IOS Press and the authors. All rights reserved

42 H. Bailer and R.D. Martin / Fama MacBeth 1973: Reproduction, extension, robustification

S II: For each time periodt−1 to t, given the risk factors betaβi and residualstandard errorssi as computed in S I, andRit the one-period percentagereturn on equityi, use subsets of

Rit = γ0t + γ1tβi + γ2tβ2i + γ3tsi + ηit, i = 1, . . . , N (2)

to produce time series estimatesγjt by means of month-to-month, cross-sectional regressions, assuming that the residual errorsη it are independentof the predictor variables.

S III: Analyze the time series of theγjt estimates using averages and t-statisticsto test the main three hypotheses C1 through C3:

C1: In any efficient portfolio, the relationship between an equities expectedreturn risk is linear, i.e.,H0 : E [γ2t] = 0.

C2: βi is a complete measure of the risk of securityi in the efficient marketportfolio, i.e.,H0 : E [γ3t] = 0.

C3: Higher risk should be associated with higher expected excess returns,i.e.,H0 : E [γ1t] = E [Rmt] − E [Rft] > 0, with Rft the risk-freerate.

FM recognized, that in S III, the use of estimatedβi in place of trueβi introducesan errors-in-the-variables problem and thatβi averaged over portfolios are moreprecise estimates of the trueβi. Therefore, FM decided to group all equities into 20portfolios based on ranked values ofβi.

However, due to estimation errors, high (low) observedβi tend to be above (below)the trueβi and portfolios formed that way tend to over (under) estimate the trueβ i.To avoid this serious regression problem, FM chose to use different data to form theβ-portfoliosβp, to estimate the initialβp-values, and to run cross-sectional regressionfor the testing period. This was done as follows:

Formation: Over a 5 or7-year formation period,βi are estimated for each stock andinitial portfolios were formed by grouping all stocks into 20 portfoliosbased on their rankedβi.

Estimation: Over subsequent 5-year estimation periods following the formationperiod, βi are re-estimated for each stock andβp re-computed byaveraging over each of the 20 portfolios.

Testing: Over another final subsequent 4-year testing period,βp were re-averaged monthly (without re-computing theβi-component) to allowfor delisting of firms, while the individualβi components were annu-ally re-computed from the beginning of the estimation period to theend of the current year of the testing period.

To test hypotheses C1, C2 and C3, FM used the following four cross-sectionalregression models (subsets of Eq. (2)):

Panel A:Rpt = γ0t + γ1tβp,t−1 + ηpt

H. Bailer and R.D. Martin / Fama MacBeth 1973: Reproduction, extension, robustification 43

Panel B:Rpt = γ0t + γ1tβp,t−1 + γ2tβ2p,t−1 + ηpt (3)

Panel C:Rpt = γ0t + γ1tβp,t−1 + γ3t¯sp,t−1 + ηpt

Panel D:Rpt = γ0t + γ1tβp,t−1 + γ2tβ2p,t−1 + γ3t

¯sp,t−1 + ηpt

wherep = 1, . . . , 20. FM’s results are summarized in FM-Table 3.The analysis in this paper will focus on Panels A and D, since Panels B and C deal

with the linearity of the single-factor market model and other sources of risk, whichare also tested in Panels A and D.

The focus of this paper is the reproducibility of the results by providing code writtenin S-PLUS [24], available upon request, and the extension of FM by applying the codeto data through 12/2002. Another focus is on the sensitivity of the OLS estimatortowards estimation error and bias when formingβ-portfolios, running cross-sectionalregressions, and averaging over month-to-month regression coefficients.

It is well known that the OLS estimate is thebest linear unbiased estimate (blue)with a convenient distribution theory for inference only when the errors areGaussian(normal distributed). Unfortunately, this is an idealized assumption that often fails inpractice. There is considerable evidence in the literature that equity returns are notnormally distributed, but rather leptokurtotic (following some heavy-tailed distribu-tion that generates outliers) and positively skewed (Cable and Holland [1]; Chou [3];Chunhachinda et al. [4]; Peiro [19]; Singleton and Wingender [22]). Outlying returnscannot only substantially bias the values of the factor beta estimates [11–13,21], butalso inflate the estimation error. The estimation error can be expressed as inflatedvariance or mean-squared-error and has been receiving increased attention in finan-cial literature [10,14,25]. However, little noticed is the fact that, while the varianceof the regression estimate goes to zero like 1/n, the bias due to non-conformingoutliers persists for arbitrarily large sample sizes. Therefore, it is also important tobe concerned about bias caused by outliers.

Martin and Simin [16] introduced to the financial community a special robustMaximum Likelihood type estimator (M-estimator) with a well-developed statisticaloptimality theory: it minimizes bias due to non-conformingoutliers while at the sametime achieving a user-specified high efficiency at the Gaussian model and maintaininghigh efficiency even at non-Gaussian, outlier generating distributions. The bias robustapproach to regression was initiated by Martin, Yohai and Zamar [18], and Gaussianefficiency-constrained bias robust solutions were obtained by Yohai and Zamar [27]and Svarc et al. [26], see Appendix A, for a brief description.

The impact of using robust methods may be washed out on a portfolio level [1,23];however, the use of the robust M-estimator may show significant differences whenapplied in S I through S III, and may also improve the regression model assumptions.Furthermore, the introduction of the M-estimator within the factor model frameworkis intended to lay the groundwork for its use in larger multi-factor models at theindividual stock returns level as in the framework of the Fama and French [7] model,or the Barra-type fundamental factor models.


Results that are based on the data in this study are referred to as replicates or BM(Bailer, Martin). The comparison with FM’s results will be restricted to regressioncoefficients, t-statistics, and coefficient of determination of FM-Table 3, Panels Aand D, and further restricted to the data presented in FM. Panels A and D will alsobe referred to as models A and D, respectively. Due to software restrictions, thenotation in graphs will be g0, g 1, g 2, g 3, and Rf in place ofγ 1, γ2, γ3, andRf ,respectively.

Section 2 describes the data. Section 3 calibrates the data to the results of FM.Section 4 extends FM through December 2002, analyzes the month-to-month timeseries averages on various time intervals, and tests conditions C1 through C3. Sec-tion 5 shows the effects of the robust M-estimator. Section 6 provides concludingcomments.

2. Data

As in FM, the data are monthly relative difference returns, adjusted for splits anddividend distributions for all stocks traded on NYSE during the time period January1926 through December 2002. The market proxy is the equally-weighted averagereturn of all stocks listed on NYSE in montht and the risk-free rate the 1-monthTreasury bill. The data was provided by the Center for Research in Security Prices(CRSP) of the University of Chicago.

3. Calibration to FM

FM-Table 1 shows the time periods and lengths of formation, estimation, andtesting periods used in S I and S II. The results in S II of FM-Table 2 are time seriesof the month-to-month cross-sectional regression coefficientsβpt, β

2pt, ¯spt, andηpt

over the testing periods from 1/1935 through 6/1968 for the 20 portfolios. S IIIproduces averages and t-statistics from these time series over various time-periods,which are summarized in FM-Table 3.

3.1. FM-Table 1

In order to be included in a portfolio, a security must be available for at least 4 yearsin the portfolio formation period, for the full 5 years in the estimation period, and inthe first month of the testing period. BM-Table 1 shows the difference between thenumber of securities meeting the data requirements in FM-Table 1 and the data used inthis study, as well as the total number of securities meeting the data requirements from1935 through 2002. The number of available securities in BM overall is somewhatsmaller than that in FM due to revisions of the CRSP database. The total number ofsecurities increased dramatically in the early Seventies when NASDAQ went live.


BM-Table 1Securities Meeting Data Requirements for FM and BM

Difference of Securities in FM and BM Total # Securities in BM

1935–1938 46 3891939–1942 22 5541943–1946 31 5761947–1950 35 6691951–1954 39 7121955–1958 34 7681959–1962 44 8121963–1966 41 8171967–1970 8181971–1974 7851975–1978 13791979–1982 13981983–1986 25321987–1990 24431991–1994 24561995–1998 29041999–2002 3109

3.2. FM-Table 2

Figure 1 (upper four panels) shows a qqplot of the values of the 20 portfoliosβpt−1

taken from FM-Table 2 versus the replicatedβpt−1. The solid line has slope one andintercept zero. The closer the points are to the solid line, the better the match. Thereplicates match FM’s results closely across all time periods.

3.3. FM-Table 3

FM-Table 3 shows the average overN month-by-month regression coefficientestimatesγj ,R2, and the corresponding t-statistics computed as

t (γj) =¯γj

s (γj)/√

N(4)

with s (·) the standard deviation over theN time periods described in FM-Table 3.Figure 1 (lower four panels) compares FM-Table 3, Panel A with replicates on 10

time-periods. The solid line has slope one and intercept zero. The closer the pointsare to the solid line, the better the replication. The replicates match the coefficientestimates, t-statistics, andR2 closely (not shown) across all time periods.

Figure 2 compares replicates with results of FM-Table 3, Panel D. The interceptestimate is not shown since its values are close to zero.

The replicates match the coefficient estimates, t-statistics, andR2 (not shown) ofFM-Table 3 reasonably well. Theγ0 shows an upward bias of the replicates, whileγ1 shows a downward bias.


1934-1938 1942-1946

1950-1954 1958-1962

g_0 t(g_0)

g_1 t(g_1)

Fig. 1. Replicates compared withβpt−1 of FM-Table 2 (upper panels) and Estimates of FM-Table 3,Panel A (lower panels).

4. Extension and tests of FM

This section extends the methodology of FM to recent time periods and tests thehypotheses C1, C2 and C3 on various time windows. The time series averages and t-statistics in S III will be taken over the full time periods from 1/1935 through 12/2002,a period excluding World War II (WWII), i.e., from 1/1947 through 12/2002, otherperiods of low and high volatility, and fourteen 5-year and seven 10-year periods,

4.1. Extension

BM-Table 1 (above) displayed the total number of securities meeting the datarequirements in this study for all testing periods from 1935 through 2002. Thenumber of firms available steeply increases in the mid-1970s and mid-1980s due toNASDAQ firms joining the market and ease in listing requirements.

BM-Table 2 gives an overview of the natural extension through 12/2002 of FMwith respect to formation, estimation, and testing periods.


BM-Table 2Time Periods in S I and S II

S I S IIPeriod Formation Estimation Testing

1 1/1926-12/1929 1/1930-12/1934 1/1935-12/19385 years 5 years 4years

2 1/1927-12/1933 1/1934-12/1938 1/1939-12/19427 years 5 years 4 years

. . . . . . . . . . . .17 1/1987-12/1993 1/1994-12/1998 1/1999-12/2002

7 years 5 years 4 years

g_1 T-STATISTICS: g_1



Fig. 2. Replicates compared with Estimates of FM-Table 3, Panel D.

4.2. Model A on selected time periods

Figure 3 shows the time series estimatesγ0 andγ1, from Model A in S II, overall testing periods from 1/1935 through 12/2002. The time series show a largenumber of outliers and different volatility regimes, e.g., the WWII period from 1935through 1940, the mid-1970s,early 1990s, and early Millennium. Therefore, classicalaverages and t-statistics will be highly dependent on the choice of time period.

BM-Table 3 shows the t-statistics for selected time periods. The t-statistic of the


BM-Table 3T-Statistics over selected Time Periods

γ0 γ1

01/1935–06/1968 2.62 2.5601/1935–12/2002 3.11 2.6901/1946–12/2002 3.14 1.9601/1946–06/1968 3.12 1.8101/1946–12/1974 2.97 1.6101/1977–12/2002

-0.2

-0.1

0.0

0.1

0.2

g_0

-0.2

0.0

0.2

0.4

0.6

1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000

g_1

Fig. 3. Time series of Regression Estimatesγ0 andγ1 from S II.

intercept stays high when volatility time periods are excluded, but the t-statistics ofthe slope parameter decreases.

Even though the power of the t-test decreases in times of higher volatility onewould still think that the t-statistics of the slope parameter could be positive andlarge, but this does not conform to BM-Table 4. Moreover, BM-Table 4 shows thatduring times of high volatility the intercept and slope tend to be insignificant.

To emphasize the dependence of the time series average and t-statistics on thechoice of time periods, the next sub-section compares time series averages and t-statistics on 5-year and 10-year periods.


BM-Table 4T-Statistics over High Volatile Time Periods

γ0 γ1

1/1935-02/1946 1.05 1.891/1974-12/1976 −0.10 1.2501/1991-03/1992 −0.05 0.4810/1999-01/2001 −0.61 1.59

-10

12

3

t(g_0)

-101

23

1935 1945 1955 1965 1975 1985 1995

t(g_1)

0.00

0.10

g_0

0.00

0.10

1935 1945 1955 1965 1975 1985 1995

g_1

-10

12

3

t(g_0)

-101

23

1935 1955 1975 1995

t(g_1)-0.0

50.1

5

g_0

-0.0

50.

15

1935 1955 1975 1995

g_1

Fig. 4. 5-Year (upper panels) and 10-Year (lower panels) Averages of Coefficient Estimates and T-Statisticsfor Model A.

4.3. Models A and D, on 5-year and 10-year Periods

Figures 4 and 5 show contiguous 5-year and 10-year averages of the month-by-month regression coefficient estimatesγj (intercept annualized) and t-statistics formodels A and D (10-year periods for model D are not shown), respectively. Figure 6shows theR2 for models A and D. On 5-year intervals, the interceptγ0 is significantfor the time periods starting in 1945, 1950, 1955, and 1985. The slopeγ 1 is neversignificant under the null-hypothesis C3, but shows a large t-statistics in the 5-yearperiod starting in 1975. Note that the null hypothesis in C3 can only be rejected whennegative values with a large t-statistics occur. The slope coefficientγ1 is negative forthe time periods starting in 1970, 1985, and 2000, but without significant t-statistics.


Fig. 5. 5-Year Averages of Coefficient Estimates and T-Statistics of Model D.

On 10-year intervals, the interceptγ0 is significant in the 10-year time periods startingin 1935 (containing WWII) and in 1945. The slopeγ 1 is never significant, but showsa large t-statistics in the 10-year period starting in 1975. The slope coefficientγ 1

is always positive. On 5-year intervals, the only significant coefficient is theγ 2

coefficient in the 5-year time period starting in 1950. Theγ 3 coefficients fluctuatewildly; however, they are insignificant on all 5-year periods.

Similar on 10-year intervals (not shown), only the quadratic termγ 2 is significanton the 10-year time periods starting right after WWII in 1945. The slope coefficientγ1 is positive except for the time periods starting in 1955 and 1995, with a larget-statistics in the 10-year periods starting during WWII. The uptrend in theR 2 since1955 with the exception of the 1980–1990 period is noticeable. This uptrend couldbe caused by market turbulences decreasing the validity of the model.

4.4. Tests of C1, C2 and C3

C1: Model D tests if the relationship between expected returns andβ is linear.With the exceptions of the 5-year and 10-year periods that include 1955, allt-statistics of theγ2 coefficient are insignificant, thus C1 cannot be rejected.

C2: Model D also tests if there is additional systematic risk besidesβ that af-fects expected returns. On 5-year and 10-year periods, the t-statistics of the


0.40

0.56

1935 1955 1975 1995

R2, MODEL D, 10 YRS

0.30

0.38

0.46

1935 1945 1955 1965 1975 1985 1995

R2, MODEL A, 10 YRS

0.40

0.60

1935 1955 1975 1995

R2, MODEL D, 5 YRS

0.25

0.45

1935 1955 1975 1995

R2 MODEL A, 5 YRS

Fig. 6. 5-Year and 10-Year Averages ofR2 for Models A and D.

coefficientγ3 are insignificant for all periods, even though the coefficientsassumed large positive values on 5-year periods. Thus, condition C2 cannotbe rejected.

C3: Model A tests for a positive trade-off between expected return and risk. Itshows that the slope coefficientγ1 is negative in the 5-year periods starting in1970, 1985, and 2000, i.e., in 3 out of 14 periods or 21.4% of the times. Itst-statistics, however, is never significant (under the null-hypothesis) and largeonly in the 5-year period starting in 1975. On 10-year periods, even thoughModel A shows allγ1 to be positive, only the t-statistics for the 10-yearperiod starting in 1975 is large. The exclusion of periods with high volatility,as shown in BM-Table 4, renders all t-statistics ofγ1 small and they remainsmall even when just periods of high volatility are considered (BM-Table 4).All taken into consideration, there is not enough evidence to reject C3 as ittests for significantγ1 � 0.

5. Impact of Robust M-Regression

It was established in Section 3, that the data and methodology of this studysufficiently replicate the results of FM. Section 4 extended FM through 12/2002and demonstrated the impact of the choice of time period on the results in S III.In this section, the extended FM will be compared with robust versions of FM.Robust versions of FM are obtained by applying the M-estimator (ROB) to variouscombinations of steps S I through S III. The efficiency of the robust M-estimatoris set to 99% throughout, providing reasonable protection against outliers, and for


MARKET RETURNS

ZIF

RE

TU

RN

S

-0.15 -0.10 -0.05 0.0 0.05

-0.2

-0.1

0.0

0.1

0.2

OLS BETA = 0.49 ( 0.14 )ROB BETA = 0.18 ( 0.13 )

Fig. 7. OLS versus ROBβ-Estimates on Monthly Returns, 1997–2002, Martin, Bailer, and Simin [17].

all practical purposes, identical behavior to the OLS estimator when no outliers arepresents, i.e., when the data is normally distributed. A basic introduction of the M-estimator can be found in Appendix. The following notation will be adapted goingforward:

RS I: S I is robustified, i.e., the OLS estimator in Eq. (1) is replaced bythe ROB estimator.

RS II: S I and S II are robustified, i.e., when the OLS estimators in Eqs (1)and (2) are replaced by the ROB estimator.

RS II ex I: S II but not S I is robustified, i.e., when only the OLS estimatorin Eq. (2) is replaced by the ROB estimator, butβ-portfolios areformed using the FM approach.

RS III: S I, S II, and S III are robustified, i.e., when the OLS estimatorsin Eqs (1) and (2) are replaced by the ROB estimator, and in S IIIthe classical averages of the time series of theγjt estimates and itst-statistics are computed robustly, again using the ROB estimator.

RS III ex I&II: Only S III is robustified by replacing the classical averages of thetime series of theγjt estimates and its t-statistics by the robustestimator, while S I and S II are done the classical FM way.

Section 5.1 motivates the potential impact when using robust methods. Section 5.2compares S I and RS I. Section 5.3 compares S II and RS II. Section 5.4 implements


Fig. 8. Month-to-Month Portfolioβp computed using OLS and ROB, 1/1935–12/2002.

the full robust approach comparing S III and RS III ex I&II and RS III. Section 5.5uses RS III in Model A and D to test conditions C1, C2 and C3. Section 5.6 looks atthe regression model assumptions.

5.1. EDA and potential impact of ROB regression

This section points out at which steps of the FM approach robust methods mayhave impact. In step S I, equities are ranked by theirβ and groups it into portfolios.Figure 7 gives an example of aβ-estimates of the Zenix Income Fund Inc. (ZIF)on the S&P500 using OLS and ROB regression. Even though the ROB regressionrejects only the single outlier large outlier in positive stock direction, the differencebetween the OLS and ROBβ is more than twice the standard error of the ROBβ.

This example indicates that the stock allocation into the 20 portfolios created inthe formation, estimation, and testing periods may well be influenced by outliers,thus producing different portfolioβp, average portfolio returns and different cross-sectional regression estimates. Figure 8 shows a qqplot of the month-to-monthportfolio βp computed using OLS and ROB. Overall the results match well withoccasional cluster of outliers as in portfolios 1, 15, 18 and 19. Furthermore, itappears that for smallerβ-portfolios, the OLSβp is downward biased, while for


Fig. 9. Time Series of Portfolio Returns used S II.

largerβ-portfolios it is upward biased. Similar results hold for the correspondingaverage portfolio returns (not shown).

In step S II cross-sectional regression estimates are computed on the portfoliolevel using month-to-monthβ-portfolio returns obtained in the S I (see Fig. 9). Thedotted lines around the center distribution are at plus and minus twice the robuststandard deviation of the returns, clearly displaying a large number of outlyingportfolio returns. The impact that ROB regression may have in S II depends on thereturns-β pairs that are identified by the ROB regression as two-dimensional outliersand, rejected in RS II. While the majority of the 816 month-to-month cross-sectionalregressions reject none or just one of the portfolio returns, there are a number ofmonths where up to 8 out of the 20 month-to-month returns were rejected (notshown). Therefore, the results in S II and RS II may well differ.

In S III, the time series of cross-sectional regression coefficients are averaged andt-statistics computed. As shown in Fig. 3 above, the time series of cross-sectionalregression coefficients, show a number of large positive and negative outliers andnon-stationary volatility. Averages taken over time periods that include negativeor positive outliers can be substantially upward or downward biased. Standarddeviations computed over time periods that include a large negative or positive outliercan be substantially inflated. T-tests formed with such quantities have little power.


g_0 t(g_0)

g_1 t(g_1)

g_0 t(g_0)

g_1 t(g_1)

Fig. 10. S I versus RS I (Upper Panels), S II versus RS II (Lower Panels) on 5-Year, 1/1935–12/2002.

Therefore the biggest difference will be expected between S III and RS III.

5.2. S I and S II versus RS I and RS II

This section compares for Model A step S I and S II with RS I and RS II,respectively, on five-year contiguous time intervals over 1/1935 through 12/2002.The comparison of the coefficients is shown in Fig. 10. The point scatter of S I versusS II is closely to the straight line with the exception of a few points corresponding tothe 5-year time periods starting in 1955 and 1990.

The match is fairly close for the scatter plot of S II versus RS III, even though theintercept is not as aligned anymore. The slope coefficient shows one outlying datapoint in the upper right corner that belongs to the 5-year period starting in 1941. TheR2 match fairly well (not shown). It also appears that the use of ROB cross-sectionalregression increases theR2, with the exception of the last time period.

5.3. S III versus RS III ex I&II and RS III

As shown in Section 5.1, time series averages and t-statistics as computed in FM-Table 3 are sensitive to outliers, and the inflated standard deviations lower the power


Table 5S III versus RS III

S III RS III5-Year γ0 t (γ0) γ0 t (γ0)

1/1935-12/1939 0.004 0.43 −0.010 −0.161/1940-12/1944 0.003 0.66 0.001 0.091/1945-12/1949 0.009 2.51 0.012 1.531/1950-12/1954 0.009 2.69 −0.009 −1.331/1955-12/1959 0.010 3.84 0.002 0.361/1960-12/1964 0.005 1.17 0.024 2.541/1965-12/1969 −0.006 −1.34 −0.016 −1.231/1970-12/1974 −0.002 −0.39 0.002 0.171/1975-12/1979 −0.001 −0.30 0.005 0.661/1980-12/1984 0.006 1.19 0.003 0.311/1985-12/1989 0.015 3.77 0.002 0.241/1990-12/1994 −0.003 −0.46 0.017 1.991/1995-12/1999 0.006 1.51 0.010 1.471/2000-12/2002 0.006 0.71 −0.003 −0.33

5-Year γ1 t (γ1) γ1 t (γ1)

1/1935-12/1939 0.013 0.82 0.006 0.081/1940-12/1944 0.016 1.79 0.020 0.871/1945-12/1949 0.001 0.22 −0.006 −0.371/1950-12/1954 0.008 1.34 0.035 2.791/1955-12/1959 0.001 0.24 0.007 0.491/1960-12/1964 0.002 0.39 −0.013 −0.591/1965-12/1969 0.013 1.82 0.022 0.851/1970-12/1974 −0.006 −0.65 −0.006 −0.181/1975-12/1979 0.023 2.57 −0.005 −0.201/1980-12/1984 0.002 0.33 0.010 0.381/1985-12/1989 −0.007 −1.22 0.062 3.081/1990-12/1994 0.010 1.23 0.049 1.941/1995-12/1999 0.008 1.12 −0.016 −1.151/2000-12/2002 −0.002 −0.12 −0.015 −0.49

of the t-tests. This section compares S III to the RS III ex I&II and RS III on 5-yearand 10-year contiguous time intervals from 1/1935 through 12/2002. Note that RSIII ex I&II means just to make step S III robust, while computing S I and S II like FM.The results are similar to implementing the full robust approach, RS III, shown inFig. 11. The results of Fig. 11 are also displayed in Tables 5 and 6. RS III lowers thepercentage of significant time periods for the interceptγ0 from 29% to 14% and raisesthe percentage of t-statistics greater than 1.96 of the slope parameterγ 1 from 7% to21% on 5-year time periods; however, on 10-year periods, it lowers the significanceof theγ0 from 29% to 14% while the percentage of largeγ1 remains even.

5.4. ROB Tests of C1, C2 and C3

C1: Table 7 shows the percentage of month-to-monthsignificant t-statistics for theFM, RS II ex I, and RS II approach from 1/1935 through 12/2002 using ModelD. The monthly t-statistics ofγ2 andγ3 in S III, Model D are significant only


Table 6S III versus RS III, 10-Year Intervals, 1935–12/2002

10-Year S III RS IIIγ0 t (γ0) γ0 t (γ0)

1/1935-12/1944 0.004 0.69 0.023 2.011/1945-12/1954 0.009 3.68 0.001 0.221/1955-12/1964 0.007 2.96 0.001 0.131/1965-12/1974 −0.004 −1.17 −0.007 −0.741/1975-12/1984 0.002 0.61 0.004 0.561/1985-12/1994 0.006 1.83 0.010 1.471/1995-12/2002 0.006 1.48 −0.002 −0.41

10-Year γ1 t (γ1) γ1 t (γ1)1/1935-12/1944 0.014 1.59 0.040 2.671/1945-12/1954 0.004 1.09 0.017 1.671/1955-12/1964 0.002 0.45 0.000 0.001/1965-12/1974 −0.004 0.63 0.008 0.401/1975-12/1984 −0.013 2.22 0.005 0.311/1985-12/1994 0.002 0.27 0.020 1.401/1995-12/2002 0.004 0.53 −0.012 −0.81

Table 7Model D: Significant t-statistics, 1/1935–12/2002

S II RS II ex I RS II

t (γ2) 18.3% 13.4% 13.9%t (γ3) 14.5% 10.3% 6.9%

Table 8Model A: T-Statistics larger than 1.96, 1/1935–12/2002

S II RS II ex I RS II

t (γ0) 49.9% 40.4% 38.6%t (γ1) 61.9% 56.5% 54.8%

t (γ1)& (γ1 > 0) 31.7% 28.6% 27.8%

in 18.3% and 14.5%, respectively, of the times. The percentage of significantt-statistics for the RS II approach is 13.9% and 6.9%, respectively. Therefore,it seems plausible not to reject C1.

C2: For similar reasons as C1, the hypothesis in C2 cannot be rejected.C3: Model A, Tables 5, 8, and 9 will provide some answers. The 5-year averages

of the slope parameterγ1 are positive and the t-statistics of C3 are neversignificant (under the null hypothesis) though values larger then 1.96 occuronly in two out of 14 time periods, i.e., in 14.3% of the time periods.

Table 8 shows the percentage of month-to-month large t-statistics for the S II, RSII ex I, and RS II approach from 1/1935 through 12/2002. Table 8 shows that thepercentage of t-statistics larger than 1.96 to be only 31.7% and 27.8% S II and RSII, respectively. Table 9 compares S III and RS III for the full time period, withand without the WWII period. In S III, the t-statistics for the interceptγ 0 is always


Table 9Model A: S III versus RS III, 1/1935–12/2002

S III RS IIIγ0 t(γ0) γ0 t(γ0)

1/1935-6/1968 0.005 2.62 0.007 1.801/1947-6/1968 0.005 3.10 0.002 0.611/1935-12/2002 0.004 3.12 0.005 1.851/1947-12/2002 0.004 3.09 0.003 0.97

γ1 t (γ1) γ1 t (γ1)1/1935-6/1968 0.008 2.56 0.006 0.691/1947-6/1968 0.006 2.09 0.009 0.981/1935-12/2002 0.006 2.69 0.005 0.811/1947-12/2002 0.005 2.08 0.006 0.98

5-YEAR: g_0 5-YEAR: t(g_0)

5-YEAR: g_1 5-YEAR: t(g_1)

10-YEAR: g_0 10-YEAR: t(g_0)

10-YEAR: g_1 10-YEAR: t(g_1)

Fig. 11. S III versus RS III, 5 and 10-Year Intervals, 1/1935–12/2002.

significant and the slope parameterγ1 always insignificant (in the sense of C3) witht-statistics larger then 1.96. The t-statistics are small in RS III. Overall, the slopeparameter is non-zero. It appears that t-statistics become insignificant when takenover shorter time periods and the robust approach tends to show smaller t-statisticsthat do not so much depend on the time-frame chosen. Keeping in mind that thedifferences between the classical and robust approach are caused by a very smallfraction of outliers, it suggests that the robust approach is more accurate. While


-0.4 -0.2 0.0 0.2 0.4

01

23

4

FMRS IRS II

Fig. 12. Model A: Probability Densities of Cross-Sectional Residuals, 1/1935–12/2002.

C3 cannot be rejected, there is not much evidence for a strong positive risk-returntrade-off.

5.5. Model check

The cross-sectional regression models in Eq. (3) assume that the error termsη pt

are serially uncorrelated and contemporaneously uncorrelated across assets, i.e.,

cov (ηqs, ηpt) = σ2p, ∀q = p, ands = t

(5)= 0, otherwise

Figure 12 shows the density estimates of the lower triangular covariance matrixfrom Eq. (5) obtained using the FM, RS I, and RS II. The model assumptions expectthe density distribution to center steeply around zero. From the three methods, RSII meets the model assumptions better then the other two approaches; still, however,the off-diagonals of the covariance matrix of the residuals are far from being zero.

6. Conclusion

The goal of this paper was to replicate the results of Fama MacBeth (1973) [9],referred to as FM, to extend the results to more recent time periods, to re-evaluate


conclusions on various time-frames, and to study the effects robust M-regression toselected stages of the three-step approach.

Although the data source was the same as in FM, database revisions and consol-idations made an exact data match impossible. However, a close match on FM’stimeframe of 1/1935 through 6/1968 was convincingly achieved.

FM’s method was extended through 12/2002 and evaluated on periods excludingtimes with much higher than usual volatility, as well as on 5-year and 10-yearcontiguous time-periods. FM’s conclusions on C1 and C2 could be confirmed andC3 not rejected; however, a conclusive positive trade-off between return and riskwas not found across all time periods considered. The results in S III depend on thechoice of time period.

The discrepancy of the results caused by outliers was strikingly confirmed inSection 5, where OLS regression and classical time series averages and t-statisticswere replaced by the robust M-estimator. The M-estimator was set at an efficiency of99% rejecting only a very small fraction of outliers. Nevertheless, the slope parameterbeta was mostly insignificant, regardless of time periods. The linear regression modelassumption, i.e., that the error terms are serially uncorrelated across assets and time,hold better using the robust M-estimator.

Acknowledgement

This research was supported by the Treasury Office at the University of Washing-ton.

Appendix A

A.1. M-Estimator

The single-factor market model Eq. (1) may be written in the general linear modelform

yi = xTi θ + εi, i =, . . . , N (A.1)

whereθT = (α, β) represents the intercept (alpha) and slope (beta) in Eq. (1) andxT

i = (1, Rmt). The class of regression M-estimates of (Huber [12]) is defined by

θ = argminθ

n∑i=1

ρ

(yi − xT

i θ

s

)(A.2)

where is a robust scale estimate to make the estimatorθ invariant with respect to thescale of the error andρ is a symmetric robust loss function. The specific loss function


OPTIMAL M-ESTIMATE RHO

x

RH

O(x

)

-4 -2 0 2 4

01

23

45

99% efficiency95% efficiency90% efficiency

OPTIMAL M-ESTIMATE PSI

x

PS

I(x)

-4 -2 0 2 4

-2-1

01

2

99% efficiency95% efficiency90% efficiency

Fig. 13. Optimalρ andψ-Functions for c-values of 0.95, 1.06, and 1.29.

ρ used in this paper was proposed by Yohai et al. [28].

ρ (r; c) =

3.25 · c2c2 ·

[1.792 − 0.972 · ( r

c

)2 + 0.423 · ( rc

)4 − 0.052 · ( rc

)6 −0.002 · ( r

c

)8]

0.5 · r2(A.3)|r/c| > 3

2 < |r/c| � 3|r/c| � 2

It is plotted in Fig. 13 for the efficiencies of 90%, 95%, and 99%, which correspond toc-values of 0.95, 1.06, and 1.29, respectively. The first order condition for optimizingEq. (A.2) with respect toθ, is

n∑i=1

xiψ

(yi − xT

i θ

s

)= 0 (A.4)

whereψ = ρ′, see also Fig. 13. Note that the OLS and LAD estimates are special casesof M-estimates, corresponding toρOLS (r) = r2 andρLAD (r) =

∣∣r2∣∣, respectively,e.g., whenc tends to infinity,ρ (r; c) becomes the fully efficientρOLS (r) = r2.


A.2. Robust T-Statistics

Let X be a sample of sizeN . Let x be the sample mean with sample variances2

and standard error ofx s.e. (x) = s/√

N . A single outlier can not only inflatex,

but alsos2 ands.e. (x), thus widening the confidence interval

[x− tn−1, α/2 · s.e. (x) , x+ tn−1, α/2 · s.e. (x)]This results in a fairly good coverageof the true value, but at the high cost of precision,or, statistically speaking, the t-test is robust with respect to type I error, but lackspower under the alternative.

The robust t-test used in this paper, obtains bothx ands.e. (x) from the robustM-regression estimate and is therefore highly robust towards outliers.

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