Time-varying parameters : an analysis on the significance and consistency of the Fama French 3-factor model

(1)

Time-Varying Parameters

An analysis on the significance and

consistency of the Fama French 3-factor

model

Kevin Tjoe Ny - 10853200

Faculteit Economie en Bedrijfskunde Econometrie en Operationele Research

Supervisor:

Mr L. (Lingwei) Kong MPhil

Bachelor Thesis

Amsterdam, June 2018 Studiejaar 2017-2018

(2)

Statement of Originality

This document is written by Student Kevin Tjoe Ny who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of com-pletion of the work, not for the contents.

(3)

Abstract

Factor models have gained popularity when performing risk analysis for portfolios, with the Fama French 3-factor model being one of the most famous. The model uses three factors namely the market, the size of a company and the value of a company. Fama and French estimated the model on a data set ranging from 1963 until 1991 and critics claim the model might have been data mined. In this paper, it is researched if the model is also significant and consistent over a larger period of time (1927 - 2017). To research the significance, t-statistics and Wald t-statistics are obtained, as well as weak identification robust t-statistics like FAR and GLS-LM. To judge the consistency of the model, the premiums and test statistics are estimated over periods of five years and displayed in graphs and tables to discover time-varying patterns. The results show that, indeed, the model is most significant and consistent over the years 1963 - 1991. The parameters in this period show lower variation and the various test statistics are the most significant. To say however that Fama and French applied data mining to obtain their results might be an overstatement.

(4)

Introduction

On first sight, using more explanatory variables, and having to estimate less parameters to select and evaluate investment portfolios may sound very appealing to investors. For years, portfolios were selected and evaluated using the Capital Asset Pricing Model (from now on referred to as CAPM) (Sharpe, 1964). However, in 1993, Fama and French in-troduced another method to explain variability in portfolios: The Fama French 3 Factor Model (Fama and French, 1993). They identify 3 factors, namely: a market premium, the size of a company and the book-to-market value (BE/ME) of a company to explain variability in portfolio returns. After the introduction of the 3 factor model, many more factors where introduced and tested. However, Bounen, de Jong and Koedijk in their paper, surveyed CFOs of 313 European firms and found out that 45% of those companies still use the CAPM model (Brounen, de Jong and Koedijk, 2004). There can be many reasons for firms to still use CAPM over the 3 factor model, but it is important to first see how Fama and French ’improved’ the CAPM model.

Under CAPM, Sharpe (1964) expresses the return of each asset by (1) the risk free rate plus (2) a risk premium. Resulting in the CAPM formula ri = rf+ βi(rm− rf). After

Sharpe, in 1993, Fama and French discover a size premium for the size of a company and a value premium for the BE/ME of a company. Fama and French assume that investment portfolios containing small sized firms will have higher returns, and investment portfolios containing low book-to-market firms tend to have lower returns. With these insights they

(6)

CHAPTER 1. INTRODUCTION

constructed a new model to evaluate portfolios, The Fama French 3 Factor Model: ri= rf + bi(rm− rf) + siSM B + hiHM L (1.1)

In this model, bi represents the market premium, si the size premium and hi the value

premium. SM B stands for ’small minus big’ regarding the size of the companies, and HM L stands for ’high minus low’ regarding the BE/ME of the companies.

It is important to question why many firms still prefer to use the CAPM model. One economist Fisher Black (1993, pp. 74-75) accused Fama and French of data mining and claims their obtained results would only hold for the data set they used. Thus, this paper is dedicated to find out if the Fama and French factors can be estimated significantly and consistently over longer periods of time or if the results only hold for the time period used by Fama and French (1963 - 1991). To research the significance, t-statistics and Wald statistics are obtained, as well as weak identification robust statistics like FAR and GLS-LM. To judge the consistency of the model, the premiums and test statistics are estimated over periods of five years and displayed in graphs and tables to discover time-varying patterns.

Before answering the research question, previous research on the model is discussed and a theoretical framework is constructed in chapter 2. In chapter 3, the used data and research methods are discussed. The results of the research are displayed and analyzed in chapter 4. Finally, in chapter 5 conclusions are drawn from this analysis.

(7)

Chapter 2

Theoretical Framework

Factor models have become very popular within finance. In order to try and see if the Fama and French model can be estimated consistently over time, more insights are given on portfolio selection and estimation, the ideas behind linear factor models and finally the Fama French 3-Factor Model itself. First, the origin of portfolio selection by Markowitz is discussed, how his ideas made people look at the risk and variance of investments and how this resulted in the CAPM. Second, the introduction of other factors is described; economist started realizing there are other priced factors next to the market premium. This ultimately led to the Fama French 3-Factor Model. Third, the previously introduced factors SMB and HML are broken down and explained in more detail. Fourth, the re-gression method (Fama Macbeth Rere-gression) is explained, after which the findings will be concluded.

2.1 Mean-Variance Optimization

The first economist to also look at the risk and variance of a portfolio was Markowitz (1952). Up until then, only the returns of assets mattered when selecting ones preferred portfolio. It is assumed investors try to maximize their own utility but for simplicity Markowitz works with a set of rules. The first one being that investors try to maximize their discounted expected returns. Second, investors see high returns as positive but high variance in these returns as negative. Especially this second rule is important. If investors

(8)

CHAPTER 2. THEORETICAL FRAMEWORK

only took in consideration the expected return of their investment there would be no diversification at all. Investors would invest all their money in the assets with the highest expected return, regardless of the risks involved (Markowitz, 1952, p. 77). Now when looking at the return on portfolios Markowitz defines ri as the return on asset i and Xi as

the fraction for asset i in a portfolio. Then the return on a portfolio consisting of assets i = 1, ..., N can be defined as:

R =

N

X

i=1

Xiri (2.1)

If then σi,j is defined as the covariance between asset i and j, and σi,i as the variance of

asset i the variance of a portfolio can be defined as:

V = N X i=1 N X j=1 σi,jXiXj (2.2)

These 2 definitions give the possibility to find a pair (R, V ) for every portfolio and thus gives the option to minimize the variance of a portfolio for any given choice of R (Markow-itz, 1952, pp. 78-81): minX N X i=1 N X j=1 σi,jXiXj st. R = N X i=1 Xiri (2.3)

2.2 CAPM

The findings of Markowitz resulted in the CAPM model. First a risk free asset is intro-duced; this asset guarantees a return of rf (Sharpe, 1964, p. 426). Now investors can

choose to hold risky asset, and a risk free asset. Sharpe states that if all investors are mean-variance-optimizers they all hold the same weights of risky assets and combine this risky portfolio with the risk free asset to get the desired combination of (R, V ) (1964, pp. 428-430). He claims that the combination of the risky portfolio and the risk free asset differs for every investor, depending on their risk aversion. In a figure with variance (σ) on the x-axis and returns (R) on the y-axis he shows one can now choose a portfolio with any return in combination with the least amount of risk resulting in the efficient frontier. This efficient frontier is a straight line passing through the points (0, rf) and (σM, rM).

(9)

and a return of rf and the point (σM, rM) indicates the market portfolio with risk σM

and return rM (1964, p. 432). Now the efficient frontier is a straight line resulting from a

linear combination of the risk free rate and the risky portfolio. The constructed efficient frontier by Sharpe (1964) can now be written as:

ri = rf +

(rM − rf)

σM

σ (2.4)

resulting in the CAPM formula:

ri− rf = βi(rM − rf) (2.5)

Where the βi of asset i is used to calculate the corresponding risk premium of the asset

(Sharpe, 1964, pp. 439-440).

2.3 Fama French 3 Factor Model

Although the CAPM model is simple to understand and implement Fama and French (2004) have shown that the model fails to work when tested empirically. What the model did do however, was inspire other economist to think about linear factor mod-els where factors other than the market premium are priced. Next to Size and BE/ME (Fama & French, 1992) as factors there is common risk in Profitability and Investment Patterns (Fama & French, 2015), Momentum (Carhart, 1997) and Earnings to Price ra-tio (E/P)(Basu, 1983). However this paper focuses on the Market factor of the CAPM model in combination with the Size and BE/ME factors resulting in the Fama French 3 Factor Model. In their paper (Fama French, 1993), they explain that next to the market premium, also the market capitalization and book-to-market value of companies is priced in their stocks. In order to test their theory they start defining the two variables SMB and HML (1993, p. 8). First they rank all the companies on the New York Stock Exchange (NYSE) on their size (Market Capitalization) and split the group at the median. They use this same median to split the stocks on the Amex and NASDAQ and thus have a group of small and big companies. Next, they split the companies into three BE/ME groups where Low represents the bottom 30 percent, Medium the middle 40 percent and High the top 30 percent. Now, six groups are created: (S/L, S/M, S/H, B/L, B/M, B/H)

(10)

where for example S/M stands for the intersect of the groups with Small size and Medium BE/ME. Now the portfolio SMB is constructed by taking the difference in returns on the three small sized company groups and the three big sized company groups. The portfolio HML is the difference in returns between the two Low BE/ME company groups, the two Medium BE/ME company groups and the two High BE/ME groups (Fama & French, 1992, pp. 8-10): SM B = 1 3(S/L + S/M + S/H) − 1 3(B/L + B/M + B/H) (2.6) and HM L = 1 2(S/L + B/L) − 1 2(S/H + B/H) (2.7) With the addition of SMB and HML to the previously defined ’market factor of CAPM, the Fama French 3 Factor Model is completed:

ri= rf + bi(rm− rf) + siSM B + hiHM L (2.8)

(11)

Fama and French (1993) test their model by splitting the companies of the NYSE, Amex and NASDAQ into 25 portfolios, where they have five groups based on their size, and 5 groups based on their BE/ME (Fama & French, 1993, pp. 21-24). For size, the rankings are: Small, 2, 3, 4, Big. And for the BE/ME the rankings are: Low, 2, 3, 4, High. If one takes the intersects of all the groups they create 25 portfolios with companies of different size and BE/ME. With data starting in June 1927 up until June 2017 it is clear the size groups of portfolios ’Small’, ’Median’ and ’Big’ has increased, however relatively to each other they experienced the same amount of growth as can be seen in figure 2.1.

2.4 Estimation

To estimate the risk premiums, Fama Macbeth Regressions (FMR) are used (Fama & Macbeth, 1973, pp. 614-615). In their paper, they explain that FMR consists of two steps, in the first step one runs a time series regression to find the β’s (the exposure to the factors) of the portfolios. This means if the data set contains i = 1, ..., n firms, for t = 1, ..., T time periods with the three Fama French factors then step one estimates.

It is first assume portfolio returns are influenced by the three Fama French factors:

Rit= αi+ βi1M RKTt+ βi2SM Bt+ βi3HM Lt+ it (2.9)

With βi,1, βi,2, βi,3 the exposures to the factors ’market’, ’SMB’ and ’HML’ for companies

i = 1, ..., n and i,t the error term. Next, they show that for step two, a cross section

regression is needed where the averages of portfolio excess returns ¯Ri are regressed on

the ˆβ’s calculated in step one to obtain the factor risk premiums with ¯Ri = _T1 PTt=1Rit.

Resulting in the following hypothesis:

H0 : Et[ ¯R] = λ0+ λ1βˆ1+ λ2βˆ2+ λ3βˆ3+ η (2.10)

2.5 Critique

Fama and French (1993) conclude their model does a good job in explaining their data since they obtain high R2 values. However, the model also got questioned by economists

(12)

like Fisher Black (1993, pp. 74-75). He explains that when a researcher tests many differ-ent model on the same data-set, he will evdiffer-entually get a model that explains the data well. In his paper he calls this data mining. The problem with data mining is that researcher never report on the models that did not fit the data well, and only publish those that do. He suspects in their paper Fama and French are affected by this problem. For example, there is no theoretical explanation for the relationship between the size and return of firms. Also for their other factor BE/ME, Black points out that Fama and French even admit that the results might be because of chance (data mining), but they find this implausible. Concluding on Black’s findings, he believes the model only works on the data set used by Fama and French in their paper in 1993. This only more so asks for an analysis outside of their data set used, which is the aim of this paper.

In this theoretical framework it is shown that Markowitz (1952) was one of the first researchers to also look also the risk of return on assets instead of only their return. With these insights he showed how to minimize the risk of achieving a certain return assuming all investors are mean-variance optimizers. From these fundamentals Fama and French were able to introduce their Fama and French 3 Factor Model that explains the returns on assets as the addition of three factors namely one for the excess market return, one for the size of a company and one for the BE/ME of a company. It is explained how Fama and French constructed their model, and how the premiums will be estimated in the remainder of this paper namely by Fama Macbeth regressions. Lastly, some earlier critique on the Fama and French 3 Factor model by Black is discussed.

(13)

Chapter 3

Research Method

When trying to find out if the parameter values remain consistent and significant over time it is important to look at the data used and discuss the methods used to answer this question. Therefore in this chapter it is first explained how the data was gathered and what different variables the data contains. Second, the method used to evaluate the Fama French factors over time are explained. Finally, multiple tests are discussed to evaluate the estimation results.

3.1 Data

For the analysis of this research, data from the library of Kenneth R. French is used. On his website, French freely provides data on the factors ’Market’, ’Small minus Big’ and ’High minus Low’ discussed in the previous chapter. He also provides returns of portfolios that are divided in 25 intersects between Size and Value of a firm. The data library starts in June of 1927 and is still updated monthly. French provides the data on daily, monthly and yearly basis. For this research the monthly returns on 25 portfolios is used ranging from June 1927 until June 2017. Also the used factor data Mrkt, SMB and HML ranges from June 1927 until June 2017 on a monthly basis.

The returns are the ’average value weighted monthly returns’ of firms in the intersect portfolios. The book-to-market value (on which the companies are sorted) is the firm value at the end of the year divided by the market value at the end of December of the

(14)

CHAPTER 3. RESEARCH METHOD

Table 3.1: Descriptive statistics portfolio returns

Portfolio Mean SE Small Low 0.57595 7.8618 2 0.69931 6.7861 3 0.99992 5.8595 4 1.1874 5.5662 High 1.3613 5.8894 2 Low 0.63818 7.0203 2 0.92573 5.8663 3 0.98732 5.2841 4 1.068 5.1703 High 1.2477 5.9383 3 Low 0.71073 6.4414 2 0.91165 5.2991 3 0.92615 4.8987 4 1.0273 4.8708 High 1.1374 5.5407 4 Low 0.7206 5.7656 2 0.75241 4.969 3 0.86201 4.8493 4 0.96842 4.7471 High 1.0391 5.6155 Big Low 0.62209 4.5705 2 0.63285 4.331 3 0.70071 4.1856 4 0.65613 4.542 High 0.95499 5.2961

prior year. It is important to use this dataset with caution. It contains financial returns from 1927 until 2017. From 1927-1956 results were recorded by writing them down, to only be digitized when computers arose. This makes the data less reliable, since it is easier for measurement errors to occur.

From table 3.1 it is clear to see that portfolios with high valued firms tend to have higher monthly returns which corresponds with the theory. It is also interesting to note that smaller sized firms have higher standard errors in their returns.

(15)

3.2 Fama MacBeth two-pass regression

As explained in the previous chapter, FMR are used to estimate the risk premiums λ. Therefore the average portfolio returns ¯Ri are regressed on the estimated ˆβi from the first

step. If X is the matrix with factors then betas are obtained in step one: ˆ

β = (X0X)−1X0R (3.1) Then the lambda obtained in step two will be:

ˆ

λ = ( ˆβ0β)ˆ −1βˆ0R¯ (3.2) Since the λs are regressed on estimated ˆβs the true variability in the λs will be higher. To account for the errors in the regressors beta a Shanken-Correction SC is applied on theˆ variance-covariance matrix of the λs (Shanken, 1992, pp.11-12). If the variance-covariance matrix of λ is defined as Ω, Shanken suggests to correct for the errors in ˆβ by (1 + c)Ω. Where c = λ0Σ−1_f λ. Resulting in a variance-covariance matrix of:

σ2(ˆλ) = ( ˆβ0β)ˆ −1βˆ0Σ−1β( ˆˆ β0β)ˆ −1(1 + λ0Σ−1_f λ) (3.3) With this corrected variance-covariance matrix it is possible to calculate the Shanken corrected t-statistics for λ. To see whether the risk premiums are significant and consistent over time, the λs are calculated for periods of 5 years. The results are plotted over time together with a confidence interval.

3.3 Test statistics

Two weak identification robust statistics are introduced and estimated in chapter 4 to test the hypothesis H0 : λF = λH0. Since a limited number of βs is used, with values

close to zero, the standard Gibbons, Ross & Shanken Wald test does not converge well, making the test less reliable (Kleibergen, 2009, p.10). However by taking the returns as R− = R1t− ι(n−1)Rnt and removing the return on the nth asset and take all other asset

returns in deviation from the nth return, together with: ˜ β = T X t=1 R−( ¯Ft+ λH0) h_XT j=1 ( ¯Fj+ λH0)( ¯Fj+ λH0) 0i _(3.4)

(16)

With ¯F matrix of demeaned factors. Kleibergen (2009, p.11) shows that statistics based on (R−− ˜βλH0) are independently distributed and have better convergence than the Wald

statistic. The moment conditions now become: E(R−_t ) = ˜βλF

cov(R−_t, Ft) = ˜βvar(Ft)

E(Ft) = µF

The F AR and GLS − LM test statistics both have a χ2 distribution that is much more reliable for testing H0 : λF = λH0 in this case where the βs obtained in the first pass lie

close to zero (Kleibergen, 2009). First, the F AR statistic is computed: F AR(λH0) = T 1 − λ0_H 0 ˆ Q(λ)λH0 (R−− ˜βλH0) 0_Σ_˜−1_(R−_{− ˜}_βλ H0) (3.5) With ˆQ(λ) = _T1 PT

t=1( ¯F + λF)( ¯F + λF). Which has a χ2(n − 1) distribution. Next, the

GLS − LM statistic is computed: GLS − LM (λH0) = T 1 − λ0_H 0 ˆ Q(λ)λH0 (R−− ˜βλH0) 0_˜ Σ−1β( ˜˜ β0Σ˜−1β)˜ −1β˜0Σ˜−1(R−− ˜βλH0) (3.6) Which has a χ2(k) distribution. Also, Shanken corrected t-statistics are compared over the 19 periods to individually test H0 : ˆλi = 0. Together with the testing of H0 : ˆλ0 = 0

the Gibbons, Ross Shanken F-statistic is computed (Gibbons, Ross & Shanken, 1989): F GRS = T − N − K N 1 + Er(f )0Ω−1Er(f ) −1 ˆ α0Σ−1αˆ (3.7) Which has an FN,T −N −K distribution.

(17)

Chapter 4

Results and Analysis

After describing the methods used to evaluate the consistency and the significance of the Fama French factors over periods of five years in chapter three, the results will be displayed and analyzed in chapter four. First the regression results of the first and second step of the Fama Macbeth two-pass regression will be discussed. Second, the graphs with plotted ˆ

λs are evaluated. Finally, obtained robust test statistics Factor AR and GLS-LM are analyzed.

4.1 Two Pass Regression

The Fama Macbeth two-pass regression is performed on each interval of five years, res-ulting in 19 different estimated Fama French models. In table 4.1 it can be seen that the first step adjusted R2s are very high. Thus the factors Mrkt, SMB and HML explain the exposure to risk quite well. However, since the constructed portfolios are value-weighted there is less noise than with individual firm returns. When using value-weighted portfolios the returns lie more closely to the returns of the market. This might explain the high R2 for some portfolios. For the Adjusted R2, the results are much lower. Later on in this chapter, it can be seen that low values for the adjusted R2 might pair with insignificant test results for FAR and GLS-LM.

(18)

CHAPTER 4. RESULTS AND ANALYSIS

Next, the obtained λs are plotted for each time period together with a coloured con-fidence interval. This will expose if certain time-varying patterns exist and if the λs are consistent and significant over time. In figure 4.1 the estimates of λ1, λ2and λ3are plotted

together with confidence intervals using α = 0.05. It is clear that λ1 is not consistent over

time, negative estimates are followed by positive ones roughly every three years. Also, zero lies in most of the obtained confidence intervals. In period 12 the estimate for λ1

is even significantly negatively different from zero, which is not what would be expected from λ1, the risk premium for market risk.

Table 4.1: Adjusted R2 for each first pass regression

Portfolio Adjusted R2 Small Low 0.69865 2 0.74608 3 0.968 4 0.89654 High 0.96447 2 Low 0.86354 2 0.96919 3 0.98396 4 0.97205 High 0.97002 3 Low 0.9194 2 0.9573 3 0.94219 4 0.96799 High 0.90761 4 Low 0.83141 2 0.79193 3 0.90183 4 0.81723 High 0.88096 Big Low 0.95714 2 0.9328 3 0.66403 4 0.81605 High 0.92326

(19)

The standard errors of λ1 are relatively large, resulting in wide confidence intervals.

For λ2 it can be seen that the estimate is mostly significant, while the sign of the estimate

varies between negative and positive a lot. The estimated λ3s are almost always positively

significant from zero. It is also interesting to note that even though the the estimate for λ3

is almost always positively significant from zero, from period 16 and onward the estimator suddenly has a very wide confidence interval, fails to be significantly different from zero and even takes on negative values. The start of period 16 equals to 2007, thus making it more interesting to note this decrease in significance of the value premium at the start of the financial crisis. This observation might ask for further analysis on how factor models perform during times of financial crisis, which brings more uncertainty and volatility. In a period where all the estimates are significant, the R2 _{of that period is also very large.}

For period three it can be seen that all three λs are significantly positive together with the fact that R2 _{equals 0.968. It is also important to again mention that data from 1927}

until 1956 should be treated with caution, since keeping track of financial data was not done by computers yet. Very large standard errors are observed especially for λ1 in the

(20)

Table 4.2: First pass R2, t(λ1,2,3), FGRS, FAR and GLS-LM

Period R2 t(λ1) t(λ2) t(λ3) F GRS F AR GLS − LM 1 0.35397 -0.74892 -3.0652 0.51504 0.26169 0.26244 0.25697 2 0.6579 -1.1043 4.4449 3.6874 0.60903 3.1665 3.1618 3 0.12845 -1.0035 -0.77813 1.2948 0.003938 0.68499 0.67898 4 0.93981 3.2588 10.578 9.2601 0.93121 60.134 60.131 5 0.6995 0.97563 -4.2818 4.6743 0.65657 0.67051 0.64642 6 0.53373 2.2054 -3.8026 0.74513 0.46712 7.7927 7.7637 7 0.47648 -2.4334 3.3162 -0.60102 0.40169 6.5823 6.5654 8 0.77856 -0.42335 1.6833 6.7762 0.74693 24.768 24.743 9 0.64938 -0.46047 4.9858 1.6291 0.59929 2.2048 2.1847 10 0.86434 1.0507 -2.8673 8.8896 0.84496 4.2788 4.2533 11 0.86932 1.3654 9.8466 -0.22888 0.85065 7.6587 7.6121 12 0.7045 -2.8219 -0.64853 4.0911 0.66229 6.4955 6.466 13 0.49345 -0.76182 -4.2766 0.24838 0.42108 0.58791 0.50298 14 0.44241 0.1848 0.99912 3.8241 0.36275 12.041 11.984 15 0.47749 -2.7633 -0.35929 2.5017 0.40285 0.10037 0.084023 16 0.8364 -3.1007 4.9685 5.6477 0.81303 19.003 18.986 17 0.35269 -1.8632 1.1778 -2.1315 0.26022 0.16618 0.13384 18 0.39202 -0.38393 -3.47 -1.0754 0.30516 2.832 2.8219 19 0.43698 0.80828 3.7254 0.66787 0.35655 -

(21)

-CHAPTER 4. RESULTS AND ANALYSIS

(a) (b)

(c)

(22)

4.2 Test statistics

In this section, the results of the previously described robust test statistics Factor AR and GLS-LM are discussed. As well as the t-statistics and GRS Wald test.

As mentioned in chapter 3, the GLS-LM statistic converges to a χ2(k) distribution. With a 5 percent confidence interval and 3 parameters this gives a critical value of 7.8147. From table 4.2 it can be seen that only in 4 time periods the hypothesis H0 : λF = 0 gets

rejected and thus all three λs significantly differ from zero. The FAR statistic converges to a χ2(n − 1) distribution. This results in a critical value of 36.41. Only one FAR statistic appears to be significant. Some GLS-LM statistics report very low values. Especially in the first 5 time periods some very low results are observed. They also correspond with very low values for the adjusted R2 in table 4.2. In period three for example, the GLS-LM statistic is 3.16, with a corresponding R2 of just 0.12. For period 16 however, the results are very significant with a high GLS-LM statistic, high (but insignificant at 5 percent) FAR statistics and a high adjusted R2 of 0.83. In figure 4.1 it can also be seen that for λ2

and λ3 the estimates in period 16 are significantly positive. Only λ1 reports a significant

negative value. An advantage of the FAR and GLS-LM statistic is that they are very robust against adjustments in portfolio returns, and still converge to a χ2 _distribution

even when the βs are close to zero in absolute value. However, this results in high critical values where under a 5 percent significance level only a few periods the zero hypothesis H0 : λF = 0 gets rejected.

If one however looks at the Shanken corrected t-statistics of λF it is clear that

espe-cially λ1 only reports two significantly positive values in period four and six. For λ2 and

λ3 the t-statistics reveal significant parameters in 7 and 9 periods respectively. For the

Gibbons, Ross and Shanken statistics it is tested if H0 : λ0 = 0. With an FN,T −N −K

distribution one obtains a critical value of 1.85. In table 4.2 it can be seen that for all the 19 periods the H0 is never rejected meaning that no λ0 is significantly different from zero:

(23)

The original research of Fama and French was conducted from 1963 until 1991. This corresponds to periods 7 to 12 within this research. Looking at the graphs in figure 4.1 it is clear that especially for λ2 and λ3 the estimates in these periods are significantly positive

with only one estimate for λ2 being negative in period 10. Also the FAR and GLS-LM

(24)

Chapter 5

Conclusion

In this paper it is analyzed whether the results Fama and French obtained in their research can be estimated significantly and consistently over a larger sample. Fisher Black accused Fama and French of data mining and thus the original model was estimated again, however on more years. First, to get a better understanding of factor models, the theory behind the Fama French 3-factor model was elaborated further after which Fama MacBeth two pass regressions were used to estimate the factor loadings ˆβ and risk premiums ˆλ. The adjusted R2 _{reported very high values for the first pass regression, possibly due to the fact}

portfolio returns are used, which follow the market more closely than individual firms. The model was estimated over periods of five years which resulted in 19 estimates of each λ. These estimates were plotted together with confidence intervals to give clear insights on their significance and consistency. Surprisingly, the estimates of λ1 were not consistent

nor significant at all. Estimates changed from being negative to positive every three peri-ods in a clear time-varying pattern and almost every confidence interval contained zero as a possible value. However, ˆλ2 and ˆλ3 resulted in far more significant results with only

a few periods of insignificance for these estimates. Especially ˆλ3 is very consistent with

only three periods of negative insignificant estimates. Interesting to note is the fact that the significance and consistency of the parameter greatly decreases after 2007, indicating different behaviour during a financial crisis. This might ask for further analysis of factor models during times of economic uncertainty and volatility.

(25)

CHAPTER 5. CONCLUSION

Also the robust test statistics FAR and GLS-LM reported many periods with insigni-ficant estimates, it was shown ˆλ1 might have played a big role in this since its Shanken

corrected t-statistics also report very insignificant results. The conducted Gibbons, Ross and Shanken test ensured that for all periods λ0 failed to be significantly different from 0.

All these findings can be combined to conclude that indeed the period over which Fama and French originally estimated their model resulted in the most significant and consist-ent results. First, comparing second pass adjusted R2 values, they are far higher during periods seven to twelve (corresponding to the years Fama and French’s original paper was based on). Second, the estimated λs were most consistent and significant in periods seven to twelve (ignoring λ1). Third, the FAR and GLS-LM test statistics are relatively

higher in periods seven to twelve with no test statistics being close to zero, while in six other period, bot FAR and GLS-LM are almost zero. Claiming Fama and French applied data mining to get the results leading to the Fama French 3-factor model is exaggerated, however it is clear the model best suits the time period they used.

(26)

Bibliography

[Basu, 1983] Basu, S. (1983). The relationship between earnings’ yield, market value and return for nyse common stocks: Further evidence. Journal of Financial Economics, 12 (1):129–156. 24

[Black, 1993] Black, F. (1993). Beta and return. The Journal of Portfolio Management, 20 (1):8–18. 24

[Brounen et al., 2004] Brounen, D., Jong, A., and Koedijk, K. (2004). Corporate finance in europe confronting theory with practice. Financial Management, 33 (4):71–101. 24

[Carhart, 1997] Carhart, M. M. (1997). On persistence in mutual fund performance. The Journal of Finance, 52 (1):57–82. 24

[Cochrane and Piazzesi, 2005] Cochrane, J. H. and Piazzesi, M. (2005). Bond risk premia. American Economic Review, 91 (1):149–173. 24

[Fama and French, 1992] Fama, E. F. and French, K. R. (1992). The crosssection of expected stock returns. The Journal of Finance, 47 (2):427–465. 24

[Fama and French, 1993] Fama, E. F. and French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of financial economics, 33 (1):3–56. 24

[Fama and French, 2004] Fama, E. F. and French, K. R. (2004). The capital asset pricing model: Theory and evidence. Journal of Economic Perspectives, 18 (3):25–46. 24

[Fama and French, 2015] Fama, E. F. and French, K. R. (2015). A five-factor asset pricing model. Journal of Financial Economics, 116 (1):1–22. 24

(27)

BIBLIOGRAPHY

[Fama and MacBeth, 1973] Fama, E. F. and MacBeth, J. D. (1973). Risk, return, and equilibrium: Empirical tests. Journal of Political Economy, 81 (3):607–636. 24

[Kleibergen, 2009] Kleibergen, F. (2009). Tests of risk premia in linear factor models. Journal of Econometrics, 149 (2):149–173. 24

[Markowitz, 1952] Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7 (1):77–91. 24

[Michael R Gibbons and Shanken, 1989] Michael R Gibbons, S. R. and Shanken, J. (1989). A test of the efficiency of a given portfolio. Econometrica, 57 (5):1121–52.

24

[Shanken, 1992] Shanken, J. (1992). On the estimation of beta-pricing models. The Review of Financial Studies, 5 (1):1–33. 24

[Sharpe, 1964] Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19 (3):425–442. 24

Time-varying parameters : an analysis on the significance and consistency of the Fama French 3-factor model