Faculty of Economics and Business, Amsterdam School of Economics Bachelor Thesis, BSc Econometrics and Operations Research
The relevance of extra factors on financial returns
An analysis of fitting and predictive power of return on equity, debt, and volatility in explaining and predicting return on assets
Dylan Houtman (11052007) June 2018
Supervisor
Mr. Lingwei Kong MPhil
Abstract
There is much debate on which is the best model to explain and predict financial returns on stocks. The Fama-French three-factor model seems to be a reliable basis. In this thesis, the Fama-French three-factor model is analyzed on possible omitted variables. Then three factors are proposed to extend this model; return on equity, debt, and volatility. The goal of this thesis is to find whether these factors are relevant in fitting and predicting returns. A three-pass regression is used to quantify the fitting and predictive power of these factors. The results show that the ROE factor is relevant for fitting the returns. Debt and volatility do not have significant power.
Contents
1 Introduction 1 2 Theoretical framework 2 2.1 Linear models . . . 2 2.2 Proposed Factors . . . 5 2.2.1 Return on equity . . . 5 2.2.2 Debt-to-equity . . . 5 2.2.3 Volatility . . . 6 2.3 Three-pass regressions . . . 6 2.4 Hypotheses . . . 8 3 Data 9 4 Method 9 4.1 Fama-MacBeth . . . 94.2 The Three-Pass Estimator . . . 10
5 Results 12
1
Introduction
Uncertainty has always played a crucial role in financial decisions. Financial decisions rely on expectations, which are never sure. In general, it is known that financial returns can be forecast by several common risk factors, such as firm size. However, the models that have been specified to forecast the financial returns differ a lot. ”A rather embarrassing gap exists between the theoretically exclusive importance of systematic ”state variables” and our complete ignorance of their identity. The comovements of asset prices suggest the presence of underlying exogenous influences, but we have not yet determined which economic variables, if any, are responsible” (Chen, Roll, & Ross, 1986, p.384). Many possible relevant economic variables have been investigated, but with different outcomes for many of them. Thus there is a debate on which additional factors do have a substantial effect on returns on investments.
In empirical asset pricing, financial returns are often modeled as a linear function of low dimensional factors. Fama and French (1993) identify three common risk factors in the returns on stocks. Following Fama and French (1993), these common risk factors can be divided into three stock-market factors: an overall market factor, a factor related to book-to-market equity and one related to the size of a firm. Fama and French (1993) conclude that these factors are a reasonable explanation for common variation in the cross-section of average returns.
This paper tries to find the relevance of factors added to the three-factor model by Fama and French. First, the factor loadings and risk premia of the Fama-French model are estimated by performing a Fama-MacBeth regression. The analysis of the estimated model indicates whether the original Fama-French model should be extended with extra factors. After this analysis, the model is estimated again, but with extra factors. This newly estimated model is estimated with three-stage regression as used by Giglio and Xiu (2017) including principal component analysis. This estimated model is used to investigate the relevance of the extra factors.
In this thesis, three extra factors are tested on relevance. The first extra factor is the return on equity. This factor could be relevant as profitability factor for predicting returns on assets. The following factor is volatility which is possibly relevant as a risk factor. When volatility for a specific asset is high, then the asset is marked as high risk. The final factor is total debt as a percentage of total capital, which is also a risk factor since high debt is associated with high risk.
This paper is organized as follows. In section 2, the theoretical framework is discussed. In the theoretical framework, the empirical background for the base models and the choice for the extra factors above is substantiated. In section 3, the data and construction of factors are discussed. Then, in section 4, the research method and tests will be explained and described. Section 5 contains the results, and an analysis of the results is given. This paper ends with section 6, the conclusion.
2
Theoretical framework
In this section, a review of literature is given. In this theoretical framework, several subjects come up for discussion. Firstly, an overview of the best-known linear models is given. Secondly, a basis and substantiation of predictive variables that proxy for extra factors in financial forecasting are given.
2.1 Linear models
The functional form is crucial for any form of modeling. Because of this, much research has been done on the underlying functional form of financial series. Traditionally, it is thought that financial series can be approached with linear regression and linear regression still is the most popular model for predicting stock-market returns that rely on economic and financial variables (Qi, 1999). Linear models are easy to interpret and can also facilely be adapted. Also, the assumptions and properties for the estimators are often well-known, so the testing of hypotheses is easier than in nonlinear models.
One of the best-known and most important models is the Random Walk Theorem. The Random Walk Theorem is based on an efficient market (Fama, 1965). An efficient market is described as a market in which all participants are educated and will try to maximize their profits with predictions based on the freely available information. However, since all investments are believed to rely on luck partly, the price will vary based on events in the past and in the future (Fama, 1965). If the stock-market is indeed efficient, this means that the prices will wander around the original value or the intrinsic value of the stock. These variations in price will be random for given uncertainty. These variations in price are the Random Walk. On the other hand, if these variations show some pattern, it is plausible to think that the future stock prices can be predicted more efficiently based on this pattern.
One of the first linear asset pricing models that try to explain the pattern in stock prices is the capital asset pricing model by Markowitz (1959). The model is as follows:
E(R) = Rf + β · (E(Rm) − Rf) (1)
With E(R) the expected return, E(Rm) the expected return on a market portfolio and Rf
the return on a portfolio without risk. This model was modified by Sharpe (1964) and Lintner (1965). They still used the same model with systematic risk β, but made it more useful in practice by stating that the non-systematic risk can be eliminated by diversification. This results in the following model:
E(RC) = α · E(RA) + (1 − α) · E(RB), with 0 < α < 1. (2)
In this model a combination of portfolios A and B is chosen, such that the ratio between non-systematic risk and expected returns is small. Now, if a risk-free rate is distracted from both sides in (1), the model can be written in terms of excess returns, which will result in the Sharpe-Lintner-Black model (Black, 1972):
These three models form the basis of how average returns and risk have been calculated and predicted for a long time. However, research has shown that the Sharpe-Lintner-Black model is misspecified. One of these misspecifications is the ’size effect’ (Banz, 1981). Banz (1981) finds that market equity has a significant effect on financial returns, especially for small firms. For average and large firms, he does not find much difference. Banz (1981) adjusted the SLB model (3) as follows:
E( ˜Ri) = ˜Rf + (E( ˜Rm) − ˜Rf) · βi+ γ2· (
φi− φm
φm
) (4)
Where γ2 is the coefficient of a variable measuring the contribution of market equity to
the expected return, φi is the market value of security i and φm is the average market value.
However, not only Banz found misspecification in the SLB model. Stattman (1980) and Rosenberg, Reid, and Lanstein (1985) found that book equity also has a significant positive relationship with the expected return on stocks. Chan, Hamao, and Lakonishok (1991) conclude that the ratio of book equity to market equity (BE/ME) has a positive influence on expected returns as well. Lastly, Bhandari (1988) states that leverage is a relevant factor in predicting returns on stocks. The relation Bhandari finds between expected returns and leverage is also positive. These misspecifications in the SLB model have been adapted by Fama and French (1992). They introduced the three-factor model including market risk, the difference in market capitalization between small and big firms and the difference in high book-to-market ratio versus small book-to-market ratio. The Fama-French three-factor model is defined as follows:
E(R) = Rf + β1· (E(Rm) − Rf) + β2· SM B + β3· HM L (5)
Where SMB stands for Small market capitalization Minus Big market capitalization and HML for High Minus Low book-to-market ratio. This model explains around 90 percent of the diversified portfolio returns. However, this model is investigated a lot, and many extra factors are added to test whether they improve the model. This results in various views on how the model can be improved or if the Fama-French model is the best.
2.2 Proposed Factors
Thus there is a debate on whether the Fama-French model is the best model for predicting or explaining financial returns. Petkova (2006) concludes that the Fama-French model is not the best model since it does not succeed in capturing the effect of significant information. To capture this effect, Petkova adds state variables to the Fama-French model. Petkova is not the only academician criticizing the Fama-French three-factor model. Many others examined the effect of extra factors on stock returns. In this thesis, three extra factors are tested for relevance in explaining and forecasting returns on assets.
2.2.1 Return on equity
The first factor is the return on equity, which is defined as net income divided by share-holders’ equity, which represents a percentage. It is a measurement of profitability by revealing how much profit is generated by the investments of shareholders. Firms with high profitability tend to have greater expected returns (Haugen & Baker, 1996). Hou, Xue, and Zhang (2014) construct the factor ROE by subtracting the average returns of the 30% smallest portfolios based on ROE from the average returns of the 30% biggest portfolios based on ROE. Hou, Xue, and Zhang find that the factor ROE earns an average return of 0,58% per month. They also find evidence that ROE is a source of common return variation missing from the Fama-French model. This could mean that ROE should be added to the Fama-Fama-French model to increase the percentage that the model can explain.
2.2.2 Debt-to-equity
The second factor proposed is the debt-to-equity ratio (DER). This ratio represents the risk of a firm’s common equity. This factor is thus a measurement of risk. An increase in the ratio increases the risk of a firm’s common equity. Moreover, the ratio of debt to equity is positively related to expected returns (Bhandari, 1988). Bhandari (1988) finds that the estimated coefficient of DER is 0,13%. This means that the factor DER earns an average return of 0,13%. Bhandari also states that the results are not sensitive to the choice of estimation
method for beta or the choice of market proxy. Johnson (2004) finds a weak unconditional positive relation between leverage (DER) and stock returns, but after controlling for firm-specific characteristics, the relation becomes negative, which is in line with research done by Penman, Richardson, and Tuna (2007). Penman, Richardson, and Tuna state that the Fama-French factors do not explain leverage results. This is interesting since the findings of Bhandari (1988) about the leverage effect would be adopted by Fama and French.
2.2.3 Volatility
The final possible relevant factor is volatility. Logically, an investor will invest if the expected return of a stock is high relative to risk. A variable for volatility represents the risk of a stock. Commonly, if a security has high volatility, it is riskier to invest in than a security with low volatility. Moreover, if volatility is high, the expected excess returns on stocks are lower than when volatility is low following Glosten, Jagannathan, and Runkle (1993). Next to Glosten, Jagannathan, and Runkle, Ang, Hodrick, Xing, and Zhang (2006) find that stocks with high sensitivities tend to have low expected returns and that firms with high idiosyncratic volatility have low average returns. Ang et al. use changes in the VIX index constructed by the Chicago Board Options Exchange to proxy for innovations. After that, they find an estimated price of volatility risk of -1% per annum. Ang et al. explain the negative sign by stating that risk-averse agents consume less when risk is high to increase savings to reduce the risk for the future because of higher uncertainty. Merton (1987) however finds the exact opposite; stocks with high firm-specific volatility have high expected returns. Merton (1987) states that this is because investors can only partly diversify away firm-specific risk.
2.3 Three-pass regressions
In general, the risk premia of factors are estimated with two-pass regressions. The Fama-MacBeth regression is an example of a two-pass regression. The two steps of this two-stage regression are as follows. First, the asset returns are regressed against one or more proposed risk factors, from where the asset’s beta per risk factor is estimated. Secondly, the cross-section
of asset returns is regressed against the estimated asset’s betas. This results in a time series of risk premia coefficients. The expected risk premium per factor is then computed by taking the average of the time series for each risk factor, to estimate the risk premia per factor. In this way, the risk premia per factor are estimated by constructing a portfolio with unit exposure to the factor of which the risk premia is estimated and zero exposure to every other factor (Fama & MacBeth, 1973).
However, this two-pass regression has potential omitted variables which would lead to biased estimators (Jagannathan & Wang, 1998). To correct for any omitted variables, all possible extra factors in the economy have to be controlled for. Most asset pricing models do not succeed in capturing all risk sources, which leads to an omitted variable bias affecting, among other things, the statistical significance. One of the often used solutions for this problem is adding arbitrary factors or firm-specific characteristics. Like this, the results are highly correlated with the arbitrary controls.
A second problem is the measurement error. The factors influencing the asset returns are often not easily measured since the details of these factors can be very complicated. Logically, this will lead to a bias in the estimations of the risk premia.
The solution to the measurement error bias and omitted variable bias proposed by Giglio and Xiu (2017) is an expansion of the generally used two-pass regression. Giglio and Xiu start with Principal Component Analysis (PCA). If a model consists of m factors, then the risk premium βt of a specific factor Ft with 0 < t ≤ m can be estimated if there are m − 1 relevant
control variables added to the model to make sure that the complete factor space is covered by the control variables and βt. However, these m − 1 control variables are often hard to
identify. PCA can solve this problem. When the number of assets in a portfolio is substantial (n → ∞), PCA can recover a set of control variables (Giglio & Xiu, 2017). This is necessary for estimating rt without the measurement error bias.
2.4 Hypotheses
The research conducted in the next part of this thesis consists of two parts. First, the basic model is analyzed. In 2.1, it is stated that the Fama-French model is a reliable basis for financial forecasting. If the Fama-French model has omitted variables, it should be extended with at least one extra variable. This leads to the first hypothesis:
• Hypothesis 1: The Fama-French three-factor model should be extended with extra factors. Secondly, the goal of this thesis is to find out whether the proposed factors have predictive power. By analyzing several extra factors, the second hypothesis can be tested:
• Hypothesis 2: The proposed factors are relevant for the Fama-French model.
In the following two sections, the data and the method of the research conducted to test the hypotheses are described.
3
Data
To construct the three new factors, monthly and yearly data of 98 different NASDAQ firms are used. These are 98 firms that have been active for at least 40 years. Although the data is often available from 1973 until 2017, the research is conducted on the data from 1994 until 2017, for which most of the data used to construct factors is available. Since most of the data used to construct the factors are yearly, a cubic spline is used to interpolate the yearly data to approximate monthly data. Then T nx2 matrices are created with column 1 the returns and column 2 the factor data of n firms at time t of T. After deleting firms with unavailable data points, these matrices are sorted on factor data. In the second last step, these T matrices are separated into two groups: low and high. Finally, the factors are created by subtracting the low average from the high average for each t in T. The data to construct the Fama-French model is from the data library of Kenneth R. French. 100 portfolios based on size and book-to-market ratio are used as R and are also constructed by Kenneth R. French.
4
Method
To test the two hypotheses, a two-stage regression and three-stage regression are performed, respectively. First, the Fama-MacBeth regression used for testing hypothesis 1 is described. Secondly, the three-stage regression is described to test hypothesis 2.
4.1 Fama-MacBeth
The Fama-MacBeth two-stage regression logically consists of two steps. In the first step, the factor loadings are estimated for each factor. In equation form, for n portfolios, all regressed on m factors:
R1,t = α1+ β1,F 1· F1,t+ β1,F 2· F2,t+ ... + β1,F m· Fm,t+ 1,t
.. .
Rn,t = α1+ βn,F 1· F1,t+ βn,F 2· F2,t+ ... + βn,F m· Fm,t+ m,t, (6)
Where Ri,t is the portfolio return of portfolio i at time t, βi,F j the factor loading of F j on
portfolio i and Fj,t the factor j at time t. Also, is assumed to be i.i.d.
In the second step, the factor loadings estimated in step 1 are used to estimate the exposure of the n portfolio returns to the m factor loadings over time. T regressions of the portfolio returns on the factor loadings are computed to achieve this:
Ri,1= γ1,0+ γ1,1· ˆβi,F 1+ γ1,2· ˆβi,F 2+ ... + γ1,m· ˆβi,F m+ i,1
.. .
Ri,T = γT ,0+ γn,1· ˆβi,F 1+ γn,2· ˆβi,F 2+ ... + γn,m· ˆβi,F m+ i,T (7)
Where R is the same as in the first step, the γ’s are the regression coefficients and the ˆ
β’s are the factor loadings estimated in step 1. These T regressions give m+1 (including the constant) series of γ for each factor. Since the assumption ∼ i.i.d. is made, risk premium γj
for factor j can be calculated by taking the average of γj over all t
4.2 The Three-Pass Estimator
In the three-pass regression, the method proposed by Giglio and Xiu (2017) is used to test hypothesis 2. First, Giglio and Xiu make two assumptions:
Assumption 1:
rt= ιnγ0+ α + βγ + βvt+ ut, ft= µ + vt with E(vt) = E(ut) = 0 and Cov(ut, vt) = 0
Assumption 2:
gt= ξ + ηvt+ zt with E(zt) = 0 and Cov(zt, vt) = 0
After these two assumptions are made, Giglio and Xiu use principal component analysis on n−1T−1R¯0R (where ¯¯ R is an nxT matrix of the returns minus the mean returns) to extract the
principal components of the portfolio returns. This is the first step of the three-pass estimator. The estimator for the factors and the corresponding factor loadings are defined as follows:
ˆ
V = T12Ξ0, with Ξ = (ξ1: ξ2: ... : ξpˆ) (8)
ˆ
β = T−1R ˆ¯V0 (9)
Where ξ’s are the eigenvectors corresponding to the highest ˆp eigenvalues of n−1T−1R¯0R¯
In the second step, Giglio and Xiu run a cross-sectional OLS regression where the returns are regressed on the factor loadings ˆβ estimated in step one. This regression is used to estimate the risk premia:
˜
Γ := ( ˜γ0, ˜γ0)0 = ((ιn: ˆβ)0(ιn: ˆβ))0(ιn: ˆβ)0¯r (10)
Where ¯r is a vector of the average returns. The final step is a time series regression where an observable factor gt is regressed on the estimated factors. This results in:
ˆ
η = ¯G ˆV0( ˆV ˆV0)−1 (11)
ˆ
G = ˆη ˆV (12)
Where G is the factor matrix. In extension to this, an analysis of the forecasting power is conducted. Using the regressions on the period 1994-2013, the returns for 2014-2016 are estimated and compared with the observed returns for this period. The approximation is defined as follows:
ˆ
R = [ι G]ˆΓi (13)
Where ˆR is the forecasted return and i is the ith column in ˆΓ corresponding to the risk premia for the ithlatent variable. In the following section, the results of conducting this method are given and analyzed.
5
Results
In this section, the empirical results are analyzed. First, the original Fama-French three-factor model is set up using the Fama-MacBeth method on the years 1994-2013, and the results are analyzed. Secondly, additional factors are added to the model and analyzed for relevance. Regression of 100 portfolios on the three factors constructed by Fama and French gives the following results shown in table 1. In table 1, the coefficients represent the risk premia γ0, .., γ3.
The Akaike’s Information Criterion for this model is -323.0378 and the Bayesian Information Criterion is -320.5200. Following the output in table 1, all risk premia are significant for predicting returns. This tells that all Fama-French factors are useful for pricing the cross-section of returns. Moreover, it tells how the factors affect marginal utility.
The adjusted R2 is notably low in comparison with the results of Fama and French. An adjusted R2 of 36,73% leaves much space for improvement. The adjusted R2 tells that the Fama-French model explains 36,73% of the variance in the cross-section. Another way to look for misspecifications in the model is by analyzing the residuals. In figure 1, a histogram of the residuals is presented. In this figure, it can be seen that the residuals seem to be normally distributed. This can also be seen in figure 2, where a quantile-quantile plot is presented. The plot appears to be linear, indicating the residuals are normally distributed. A Jarque-Bera test for normality gives a p-value of 0,0198, which will reject the null hypothesis. Thus there is enough statistical evidence for normality. Also, a Durbin-Watson test for autocorrelation gives a p-value of 0,1147, not rejecting the null hypothesis that the residuals are uncorrelated.
Normality of residuals does not mean there is no left-over structure in the residuals. When the average returns are plotted against the residuals, there appears to be a linear structure in the residuals. In figure 3 the plot of average returns against residuals is shown. It is clear that there is a pattern in the residuals and the plot is not a random cloud. This structure could be the results of omitted variables. This gives reason to assume that the Fama-French factors do not succeed in capturing the complete cross-section of returns.
If this regression is conducted with the proposed extra factors return on equity, debt, and volatility, the results are as shown in table 2. The Wald test of this model in comparison with
the Fama-French three-factor model gives a p-value of 0.0371, which tells that the model with six factors has a better fit than the Fama-French three-factor model. The Akaike’s Information Criterion is -325.0535 and the Bayesian Information Criterion is -329.5016. AIC for the ex-tended model is lower than in the original model, implying model improvement. However, it is a weak result since the difference is minimal. On the other hand, The ∆BIC is approximately 9, which is substantial evidence that the extended model is better. It is also interesting to look at the adjusted R2 which has increased to 40,35%. This increase of 3,62% raises the question whether the goodness of fit in this model increased because of the power of the extra factors or because of noise in one or more extra factors. The only factor that has a risk premium sig-nificantly different from zero is the return on equity. This tells that ROE is useful for pricing the cross-section of returns. It also tells that ROE affects marginal utility negatively. This is in line with the economic theory. When return on equity increases, risk decreases. Debt and volatility do not seem to be relevant for pricing the cross-section. However, it is possible that debt and volatility are still useful.
When the proposed factors are added separately, the results are as shown in table 3. These results are in line with the results in table 2. The adjusted R2 of the regression including ROE is the only adjusted R2that increases after adding the proposed factor. Moreover, the adjusted R2’s of regression with debt and with volatility are lower than in regression with the Fama-French factors only. Also, Wald tests with each regression in table 3 and the Fama-Fama-French model give probability values of 0.0066 for the model with ROE, 0.8569 for the model with debt and 0.5286 for the model with volatility. Thus these Wald tests only reject the null hypothesis that the Fama-French three-factor model is as good as the expanded model for ROE. The AIC of the three regressions supports the results found by analysis of separate addition of the factors. For ROE, debt, and volatility, the AIC’s are -328.4949, -320.8010, and -321.2486 respectively. Only the AIC for regression with ROE is lower than the AIC for the original model by Fama and French. The AIC’s for regression with debt and volatility do not imply model improvement. The result for ROE is supported by a BIC of -328.2481 giving strong evidence for improvement (∆BIC ≈ 7,7). The results for debt and volatility are not directly supported, but the BIC’s of respectively -320.5541 (∆BIC ≈ 0.0) and -321.0017 (∆BIC ≈ 0.5) are not worth anything but
a mention.
To investigate the power of the factors, the method of Giglio and Xiu (2017) is used to estimate the risk premia with PCA. The R2 of the time series regression conducted as the third step of this method shows the signal-to-noise ratio per factor as shown in table 4. Each column represents the signal-to-noise ratio per latent variable. The latent variables correspond to the largest four eigenvectors in n−1T−1R¯0R. Following the results in table 4, the three¯ Fama-French factors have relatively high signal-to-noise ratios in comparison with the three proposed factors. The signal-to-noise ratios of the proposed factors are quite low, indicating that they consist mainly of noise. This means that the power of the three proposed factors is quite low. For ROE, this could mean that the factor is a proxy for a true underlying factor since ROE itself is relevant for pricing the cross-section. For debt and volatility, it supports the insignificance shown in table 2. Debt and volatility are most likely redundant.
The proposed factors seem to fit the data mediocre. The profitability factor return on equity is significantly different from zero. However, all three extra factors seem to consist mainly of noise. All factors seem to have only little predictive power when looking at the signal-to-noise ratio. These are results when looking at the results for 1994-2013. To extend the investigation of the predictive power of the model, the estimated returns for 2014-2016 are compared with the actual returns. The results of the regression on 1994-2013 are used to find an approximation for ˆR. This approximation is plotted against the average observed returns.
In figure 1, month number 1 represents January of 2014, and month number 36 represents December of 2016. Looking at figure 1, it is interesting that the observed returns roughly seem to follow the same path as the estimated returns with latent variable 3 and 4 but with a two-month lag. This could imply that the factors do have predictive power for the returns two months ahead. As can be seen in figure 2, the lagged estimated returns cover the observed returns roughly. In general, the observed returns increase when the estimated returns increase. However, the actual returns fluctuate a lot more, which means that the six factors jointly only capture the fluctuations in returns partly.
6
Conclusion
The goal of this thesis is to find the relevance of adding several extra factors to the well-known Fama-French three-factor model. A Fama-MacBeth regression is first used to investigate if the Fama-French factor model should be extended with extra factors. Secondly, a three-pass regression as proposed by Giglio and Xiu is used to find the relevance of three possibly significant factors; Return on equity, debt, and volatility. It is found that the adjusted R2 of
the Fama-French model leaves room for improvement and there is a left-over structure in the residuals. After adding the extra factors, the adjusted R2 improves, and several tests show that the extended model is better.
However, only ROE is relevant for explaining the cross-section of returns. In contrast to the probability factor, the risk factors are not significant. Moreover, all proposed factors have a low signal-to-noise ratio, a measurement of predictive power. The investigation of predictive power continues with a comparison of the observed and predicted returns. This analysis leads to two months lagged factors. The conclusion is that all proposed factors do not have enough predictive power. The profitability factor can be useful in explaining the returns. For forecasting, further research can be conducted on lagged terms. The Fama-French model seems to ask for an extension and ROE seems to have significant explaining power. However, all proposed factors lack predictive power.
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Appendix
Table 1: results Fama-MacBeth regression with Fama-French factors
Table 3: Results Fama-MacBeth regressions with Fama-French factors and each proposed factor
Figure 1: Histogram of the residuals in Fama-French three-factor model
Figure 3: Plot of the average returns on residuals
