Power distortion in tests of risk premia in linear
factor models
Thesis
July 3, 2015
Universiteit van Amsterdam
Author
: Sean Tervooren 6170242Contents
1 Introduction 1
2 Factor model for portfolio returns 2
2.1 Factor models with unobserved factors . . . 4 2.2 Factor models with observed proxy factor . . . 4
3 Data analysis 6
4 The FM Two Pass Procedure 11
4.1 Simulating the R2
OLS for the observed proxy factors . . . 13
5 Tests of risk premia 17
5.1 Power and size simulation . . . 21
6 Testing methods for a selection of risk premia 24
6.1 Investigation of the risk premia for the Kenneth French data set 25
1 Introduction
Linear factor models are very widely used in nance as shown by Fama and French (1973). This resulted in a very well known nancial model, which is called the capital asset pricing model (CAPM) and implicates a linear factor structure for asset returns, Lettau (2001). These models are concerned with the relationship between portfolio returns and economic factors. This relationship is often investigated by using the Fama and MacBeth (FM) two pass procedure as shown by Fama and MacBeth (1973) and Cochrane (2005). The rst pass procedure estimates the β0s of the factors using a linear model. The second
pass procedure regresses the average portfolio returns on the β0s to estimate
the risk premia. The R2 of the second pass serves as a measure of strength of
the relationship between the portfolio returns and the observed factors. But in linear regression models the β0sare sensitive to collinearity of the explanatory
variables, this means that the risk premia are sensitive to the collinearity of the β0s. Collinearity occurs when the β0sare close to zero and can lead to problems with hypotheses tests.
There is a vast amount of literature available on suitable factors that can be used in the linear factor model. Fama and French (1992) suggests ve dier-ent factors that can be incorporated in the factor model. These include excess market return and average return of high book to market equity ratio compa-nies minus the average return of low book to market equity ratio compacompa-nies and average returns of small stocks minus average returns of big stocks within a port-folio. Jagganathan and Wang (1996,1998) suggests the factors of consumption growth and labor income growth because these re ect economic development or downturn.
Observed proxy factors are used to capture the unobserved factor structure in the portfolio returns. When these observed factors provide an accurate proxy for the unobserved factors the β0s of the rst step are spanned by the β0s of
the true factors and the R2
OLS is large, see Lewellen (2010). To thoroughly
investigate the accuracy of the R2
OLS one should investigate the unexplained
factor structure of the rst step residuals. This is done by investigating the principal components as shown by Kleibergen (2014) and Jollie (2002). When this unexplained factor structure is large the OLS R2 is not an indication of
the strength of the relationship between the expected portfolio returns and the observed proxy factors. Kleibergen (2014) shows that adding useless factors can increase the R2
OLS but does not change the unexplained factor structure of the
rst step residuals.
Small β0s are a result of collinearity of the explanatory variables. Even
for size-able β0s the risk premia estimates can be misleading if the number
of assets is large. The risk premia obtained in those situations are usually too small and their con dence sets are not precise. This incorrect precision of the con dence sets shows that statistical inference based on Wald t-statistics is misguiding and that the large sample t-distribution diers from the normal one. Kleibergen (2006) suggests test statistics whose large sample distribution remain unaected by the small values of the rst step β0s. These statistics are
therefore more reliable than the commonly used Wald and t-test statistics when estimating a con dence set of the risk premia in a linear factor model.
To estimate the risk premia a correct model has to be formed. This means incorporating the right observed proxy factors in the model. These models can be found by investigating which of the suggested observed factors reduce the unexplained factor structure of the rst step residuals. It is also important to show what the eect is of adding useless factors on the R2
OLS and on the
unexplained factor structure. When the correct factor model is speci ed and the risk premia are estimated it is vital to test these estimates. Because of the threat of small β0s it is important to select the correct test statistic. But how
do the test statistics behave when the β0s are small? This behavior can be
simulated using a Monte Carlo simulation.
This research starts in section two by explaining the factor model theory and the performance measures that are used to estimate the strength of the model. Secondly the data is analyzed and the suggested proxy factors are tested. The eect of useless factors on the R2
OLS is simulated in section four. In section ve
the power of the test statistics is simulated and in section six the risk premia of the speci ed model are tested. This research ends with a conclusion in section seven.
2 Factor model for portfolio returns
The rst step is to describe the relationship between portfolio returns and eco-nomic factors. The portfolio returns can be described in a factor structure with
k factors as shown by Kleibergen (2014):
rit= µRi+ βi1f1t+ ... + βikfkt+ εit, i =, ..., N, t = 1, ...T ; (2.1)
with rit the return of the i-th portfolio in period t; µRi the mean return
on the i-th portfolio; fjt the realization of the j-th factor in period t; βij the
corresponding factor loading of the j-th factor for the i-th portfolio, εit is the
disturbance for the i-th portfolio return in time period t and N is the number of portfolios and T is the number of time periods. The factor model can also be described in in vector notation:
Rt= µR+ βFt+ εt, (2.2) with Rt= (r1t...rnt)0, µR= (µR1...µRN)0, εt= (ε1t...εN t)0 and β = β1t · · · β1k ... ... ... βN 1 · · · βN k , (2.3)
The vector notation of the factor model in equation 2.2 shows that the factors are i.i.d with nite variance and are uncorrelated with the disturbances εt, which
are i.i.d and have nite variance as well. This means that the covariance of the portfolio returns has the following equation:
VRR= βVF Fβ0+ Vεε, (2.4)
with VRR the N"N dimensional covariance matrix of the portfolios, VF F the
k"k dimensional covariance matrix of the factors and Vεεis the N"N dimensional
covariance matrix of the disturbances.
The dierent factors that are incorporated in the model aect all portfolios simultaneously. With the use of principal component analysis it is possible to investigate how many factors play a vital role in the portfolio covariances. Principal component analysis uses a spectral decomposition of the covariance matrix VRR of the portfolio returns:
VRR= P ΛP0, (2.5)
with P = (p1...pN) a N"N orthogonal matrix of the eigenvectors of VRR
and Λ a N"N diagonal matrix of the eigenvalues of VRR with the eigenvalues
in descending order on the diagonal of the matrix. The number of factors can be estimated by investigating the number of eigenvalues that are signi cantly larger than the other eigenvalues.
2.1 Factor models with unobserved factors
Kleibergen (2014) uses a measure to check for the presence of a factor structure. This measure estimates the fraction of the total variation that is explained by the largest three principal components. The measure is speci ed as follows:
F ACCHECK = λ1+ λ2+ λ3
λ1+ ... + λN
, (2.6)
with λ1> λ2> ... > λN the eigenvalues of equation 1.5 in descending order.
This factor structure check shows which percentage of the data is explained by the largest three principal components. Principal component analysis is used to reduce the number of factors in the unexplained factor structure of the residuals. This method is used when there is no preliminary knowledge of the factors in the model.
2.2 Factor models with observed proxy factor
There exists a vast amount of literature which explains portfolio returns using observed factors who are proxy for unobserved factors. The observed proxy factors consist of nancial based factors and macro economic factors. The ob-served model is similar to the model in equation 2.2 only the value of Ftis now
observed and the number of factors are known, lets say there are m factors:
with Gt = (g1t...gmt)0 the m-dimensional vector of proxy factors, Ut =
(u1t...umt)0 a N-dimensional vector of disturbances, µ a m-dimensional vector
of constants and B the N" m dimensional matrix that contains the β
0sof the
proxy factors a liated with the portfolio returns.
The eigenvalues of the covariance matrix can be used to test the signi cance of the parameters of the proxy factors. Kleibergen (2014) suggests a likelihood ratio (LR) test statistic for testing the null hypothesis that the parameters of the proxy factors are equal to zero. This leads to the hypothesis H0 : B = 0
against the alternative hypothesis that the proxy factor parameters are unequal to zero, H1: B 6= 0. The LR test equals
LR = T [log(| ˆVport|) − log(| ˆVres|)] = T N
X
i=1
[log(λi,port) − log(λi,res)], (2.8)
with ˆVport and ˆVres estimators of the covariance matrix of the portfolio
returns and the residual covariance matrix after regressing the proxy factor on the portfolio returns, and λi,port,i = 1, ..., N, the eigenvalues of the portfolio
returns covariance matrix, ˆVport, and λi,res,i = 1, ..., N , the eigenvalues of the
covariance matrix of the residuals of the observed factor model, ˆVres. The LR
statistic has a χ2(3N )distribution in large samples.
Another measure to investigate the correctness of the factor model is the pseudo-R2. This is a goodness of t measure that investigates the percentage of
the total variation of the portfolio that is explained by the proxy factors. The total variation of the portfolio returns is measured by the sum of the eigenvalues of the covariance matrix and the total variation of the proxy factor is calculated in the same way. Because the total variation of the observed proxy factors are equal to the total variation of the portfolio minus the total variation of the residuals of the regression of the portfolio returns on the observed proxy factors, the pseudo-R2follows the following formula:
pseudo − R2= 1 − PN i=1λi,res PN i=1λi,port (2.9)
3 Data analysis
In order to investigate the number of principle components in the data, this re-port uses data acquired from the Kenneth French website. The website consists of multiple portfolios which are formed out of stocks from the NYSE, AMEX and NASDAQ, of which market equity data and (positive) book equity data is available from the rst of July of 1926 to the 27-th of February of 2015. Because this time period contains a large amount of observations and extreme outliers, which have an signi cant eect on the estimations, this research limits the data set to the 650 most recent observations.
This report uses several data sets consisting of twenty- ve portfolios to in-vestigate the number of principal components in these data sets. The rst data set that is used consists of portfolio's which are selected on size (market equity, ME) and book equity to market equity ratio (BE/ME). The data set consists of twenty ve portfolios which are an intersect between ve portfolio's selected on ME and ve portfolio's selected on BE/ME. The breakpoints for the selec-tion criteria are NYSE quintiles. The second data set consists of twenty- ve portfolio's selected on size and on investment (Inv). Investment is the change in total assets from the scal year ending in year t-2 to the scal year ending in t-1, divided by t-2 total assets. The investment and size breakpoints are NYSE quintiles. The nal data set consists of twenty- ve portfolios which are the intersect of ve portfolio's selected on size and ve portfolio's selected on operating pro tability (OP). Operating pro tability for June of year t is annual revenues minus cost of goods sold, interest expense, and selling, general, and administrative expenses divided by book equity for the last scal year end in t-1. The OP and size breakpoints are NYSE quintiles.
1920 1940 1960 1980 2000 2020 −40 −20 0 20 40 60 80 100 Date Return
Small size firms portfolio returns
Small Growth Small Neutral Small Value 1920 1940 1960 1980 2000 2020 −40 −20 0 20 40 60 80 Date Return
Big size firms portfolio returns
Big Growth Big Neutral Big Value
Figure 1: Returns for six portfolios. On the left the portfolios of the small size companies and on the right the portfolios of the big size companies.
Figure one shows the returns of six portfolios formed on size and book to market ratio. The gure on the left shows the portfolio returns of small size companies and the right hand gure shows the portfolio returns of big size companies. The gure shows that the portfolio returns of the big size companies are less volatile then the returns of the small size companies. Interestingly the graph shows large volatility peaks just before the First World War, the Second World War and the credit crunch in 2008. The rst two are periods of extremely high volatility and therefore greatly in uence the estimation results. As a consequence this research leaves these two periods out of the data set as is described at the beginning of this section. Both graphs show that value rms, these are rms with high book to market ratio, are in general more volatile then neutral or growth rms.
This research uses two sets of parameters to obtain the LR and the pseudo-R2 tests.The rst set of factors used in the factor model are the three observed
proxy factors of the paper of Fama (1992). This paper suggests a factor model with three factors which consist of excess market return (Rm− Rf), book equity
to market equity ratio (BE/ME) and size (ME) as factors in the model, this means that m is equal to three. These observed proxy factors are also available on the Kenneth French website. The proxy factor for book equity to market equity ratio is the average returns of rms with high BE/ME minus the average return of rms with low BE/ME (HML). The proxy factor for ME is the average returns of two rms with small ME minus the average returns of rms with big ME (SMB). The second set of factors is formed by the three factors of the rst factor set extended with the Robust Minus Weak (RMW) factor. RMW is the
average return on the two robust operating pro tability portfolios minus the average return on the two weak operating pro tability portfolios. The factor set is also extended with the Conservative Minus Aggressive (CMA) factor. This is the average return on the two conservative investment portfolios minus the average return on the two aggressive investment portfolios. Therefore the second factor set consists of ve factors which are all available on the Kenneth French website. For both the portfolio returns and the factors the number of time observations is T=650.
25 size and book to market sorted portfolios Vport Veeˆ 3 factors 5 factors 1 2732,23 944,12 933,07 2 959,85 310,37 299,68 3 325,71 187,71 187,23 4 205,19 68,44 60,51 5 173,72 39,67 35,27 6 106,63 31,22 29,28 7 38,10 23,51 22,58 8 28,73 20,84 20,59 9 23,58 19,23 17,64 10 20,21 15,78 15,52 FAC 3 factors 0,79 0,68 0,68 FAC 5 factors 0,86 0,73 0,73 LR 387,63 429,43 pseudo-R2 0,30 0,33
(a) Portfolio's sorted on size and Book-to-market
25 size and investment sorted portfolios
Vport Veeˆ 3 factors 5 factors 1 2742,68 722,38 719,22 2 731,48 54,78 50,14 3 146,65 32,41 27,95 4 73,58 29,40 25,30 5 34,14 27,08 21,34 6 27,74 25,39 17,52 7 25,72 17,57 15,61 8 18,04 15,82 14,67 9 16,75 14,60 14,38 10 15,07 13,40 12,94 FAC 3 factors 0,84 0,58 0,59 FAC 5 factors 0,87 0,62 0,63 LR 805,84 857,74 pseudo-R2 0,53 0,55
(b) Portfolio's sorted on size and investment
25 size and operating pro tability sorted portfolios
Vport Veeˆ 3 factors 5 factors 1 2744,94 221,73 219,38 2 225,31 209,52 208,00 3 211,31 90,73 53,06 4 128,12 51,03 31,69 5 86,12 31,43 24,71 6 30,80 20,43 19,70 7 25,08 18,96 16,74 8 20,32 16,91 15,15 9 18,36 15,19 13,33 10 16,07 13,08 13,08 FAC 3 factors 0,80 0,46 0,45 FAC 5 factors 0,85 0,53 0,51 LR 993,73 1108,96 pseudo-R2 0,60 0,64
(c) Portfolio's sorted on size and pro tability
Table 1: FACCHECK, LR and pseudo-R2 for various portfolio's
Table one shows the ten largest eigenvalues of the covariance matrix of the portfolio returns and of the covariance matrix of the residuals after using the two
observed proxy factors parameter sets. Table 1a uses the portfolio's which are selected on size and book to market ratio. The table shows that the parameter set with the three proxy factors reduces the factor structure from six large components to three. The set with the additional proxy factors does not reduce the factor structure any further. This therefore indicates that the two additional observed proxy factors are not useful. Looking at the LR test and the pseudo-R2 tests shows an interesting result. The value of the LR test when using
ve factors is higher, but the unexplained factor structure of the residuals is not reduced. The pseudo-R2 test shows that when increasing the number of
parameters the test statistic increases from 0.305 to 0.332. This means that the percentage of total variation that is explained by the proxy factors does increase when the number of proxy factors is increased. This pattern is visible in all three of the data sets. Increasing the number of proxy factors increases the percentage of total variation that is explained. The pseudo-R2 test also
shows that for the data set with the portfolio's sorted on size and operating pro tability the explained percentage of total variation is highest. It is followed by the data set with the portfolios selected on size and investment. But the unexplained factor structure of the residuals remains the same when adding the two additional proxy factors. This contradicts the results shown by the LR-test and the pseudo-R2tests.
It seems that for all three data sets the three factors are the best t. Al-though adding two extra factors improves the pseudo-R2 test, this does not
reduce the unexplained factor structure of the residuals which means they do not supply any additional information.
Table two shows the ten largest eigenvalues of the covariance matrix of the portfolio returns and of the covariance matrix of the residuals after using the three Fama and French (1992) suggested factors and some additional factors. Jagganathan and Wang (1996,1998) suggested some additional proxy factors. They suggested that consumption growth and labor income growth are good indicators of economic growth or downturn. These are combined with the three previous factors and the unexplained factor structure of the residuals is investi-gated. For this investigation I used twenty- ve size and book to market sorted portfolios, this number of portfolios is optimal because the matrix of the port-folio returns has to be inverted in a later stage in this research and this will require a lot of calculation capacity. The table shows that labor income growth and consumption growth do not reduce the unexplained factor structure of the residuals. This means that these do not supply any additional information and
are therefore not suitable to act as observed proxy factors.
25 size and book to market sorted portfolio's
Vport Vˆee
3 factors 5 factors (consumption and income)
1 620,1223 17,3994 17,8140 2 26,0358 6,1866 6,2262 3 10,3797 4,3503 4,3624 4 5,9554 3,6342 3,7053 5 4,3955 3,0544 3,0620 6 3,4906 2,6541 2,6895 7 3,1484 2,5413 2,5701 8 2,7047 2,3368 2,4192 9 2,5392 1,9186 1,91856 10 2,3615 1,8049 1,8085
Table 2: Factor structure when using consumption growth and labor income growth
4 The FM Two Pass Procedure
The next step is to calculate the risk premia, which indicates the relationship between the three selected factors and the portfolio returns. This requires ad-ditional information on factor models and an explanation of the FM two pass procedure.
According to Cochrane (2001) and Kleibergen (2013) stochastic discount factor models show a relationship between the expected returns on the portfolios and the β0sof the portfolio returns with their (unobserved) factors:
E(Rt) = ιnλ0+ βλF (4.1)
with ιn the N-dimensional vector of ones, λ0 the zero β return and λF the
k-dimensional vector of risk premia. Fama and Macbeth (1973) propose a two pass procedure to estimate the risk premia:
1. Estimate the observed factor model in equation (1.7) by regressing the portfolio returns Rton the observed proxy factors Gtby least squares to obtain
ˆ β = T X t=1 ¯ RtG¯t 0 ( T X t=1 ¯ GtG¯t 0 )−1 (4.2) with ¯Gt= Gt− ¯G, ¯G =T1PTt=1Gt, ¯Rt= Rt− ¯Rand ¯R = T1PTt=1Rt.
2. Regress the average returns on the vector of constants ιn and the
es-timated B, to obtain estimates of the zero-β return λ0 and the risk premia
λF : ˆ λ0 ˆ λF ! = " ιn... ˆB 0 ιn... ˆB #−1 ιn... ˆB 0 ¯ R. (4.3)
The FM two pass procedure uses the least squares estimator that results from the observed factor model to estimate the risk premia and the zero-β return. The quality of the estimates of the two pass procedure depends on the ability of the factor model to capture the factor structure of the portfolio returns. This is explained by the following linear regression between the unobserved factors Ft
and the observed proxy factors Gt:
Ft= µF+ δGt+ Vt, δ = VF GVGG−1 (4.4)
with VF Gthe covariance between the unobserved and observed proxy factors,
and VGGthe covariance matrix of the observed proxy factors, and Vtand Gtare
assumed to be uncorrelated with t since Ft is uncorrelated with t. Inserting
equation (3.4) in (1.2) we obtain the following equation for the portfolio returns: Rt= µR+ βµF+ βδGt+ βVt+ t= µ + βδGt+ Ut (4.5)
with µ = µR+ βµF, Ut= βVt+ t.The δ is small or zero when the observed
proxy factors do not explain the unobserved factors very well and Vtis large and
proportional to the unobserved factor Ft. A large Vt suggests an unexplained
factor structure in the residuals of Ut of the observed factor model because
Ut = βVt+ t. A small value of δ also implies the estimated ˆB of the two
pass procedure is very small because equation 3.5 shows this equals βδ. The results for the FM two pass procedure are derived under the assumption that the estimate of ˆB is a full rank matrix so
ˆ
B→ βδ,p (4.6)
is a full rank matrix, as shown by Fama and Macbeth (1993).
One of the commonly used statistics in the FM two pass procedure is the OLS R2. The R2is a goodness of t measure that is commonly used to measure
the explanatory power of a regression.
OLS R2.
The OLS R2is the explained sum of squares divided by the total sum of squares
when we only use a constant term so its expression reads R2OLS= ¯ R0PMιnR¯ ¯ R0M ιnR¯ = ¯ R0M ιnB( ˆˆ B 0M ιnB)ˆ −1Bˆ0M ιnR¯ ¯ R0M ιnR¯ (4.7) with PX = X(X0X)−1X0, MX = IN−PXfor a full rank matrix X and INthe
N"N dimensional identity matrix. The R2
OLS is analyzed under the assumption
that the observed and unobserved factors are only minorly correlated.
4.1 Simulating the R
2OLS
for the observed proxy factors
The following simulation describes the properties of the R2
OLS. This simulation
is comparable to the simulation results of Kleibergen (2013). The FM two pass procedure is used to estimate the risk premia on the three Farma and French (FF) factors using the returns on twenty- ve size and book to market portfolios from 1978 to 2015. The portfolio returns are generated by the factor model in equation (1.2), with µ = ιnλ0+ βλF, and E(Ft) = 0 using the estimated
values of β, λ0 and λF as true values and factors Ftand matrices ˆVF F and ˆV
with ˆVF F the covariance matrix of the three FF factors and ˆV the residual
covariance matrix that results form regressing the portfolio returns on the three FF factors. The number of time series observations is 421.
The rst panel uses the simulated portfolio returns to estimate the density function of the R2
OLSin equation (3.7) using an observed factor Gtthat initially
contains only one of the FF observed proxy factors, then the rst two factors and then all three factors. Figure 2 shows the resulting density functions of R2
that the density function of the R2
OLS moves to the right when an additional
true factor is added. Figure 2 shows that R2
OLS is close to one when all three
true factors are added. The gure also shows to what extent the observed factor model explains the factor structure of the portfolio returns. When the simulation only uses one observed factor the largest three eigenvalues explain around 80% of the variation. When two factors are used this drops down to 60% and when all three observed factor are used the largest three eigenvalues explain 45% of the variation.
Figure 3 shows the density function of another simulation experiment. In this simulation the same model is simulated only in contrast to the previous experiment this simulation estimates an observed factor model with only useless factors. The simulations starts with only one useless factor and moves on to two and three useless factors. The useless factors are generated by a normal random distribution which is centered around zero and has a standard deviation of one.
The density functions of R2
OLSin gure 3 are higher than the density function
of R2
OLSwhen only one true factor is used. This implies that, when performance
is based on the R2
OLS, observed factor models with useless factors outperform
models with only one true factor model. It also shows that the observed factor model with three useless factors outperforms the observed factor models with two true factors because the R2
OLS of the former outperforms the R 2
OLS of the
latter. Kleibergen (2013) states that this becomes even more apparent when one adds even more useless factors. The FACCHECK simulation shows that the useless factors actually explain nothing in the observed factor model. The density of all three factor combination are all similar and show that the three largest eigenvalues explain 92%, meaning that these factors matter vary little.
The results of gure 4 show similarities with former simulations. This simula-tion starts with one true factor and continues with adding one and two irrelevant factors. Figure 4 shows, which is similar to gure 3, that adding irrelevant fac-tors increases the density of R2
OLS. This is also con rmed by the FACCHECK
simulation. The useless factors do not explain anything. The dierence with the FACCHECK of gure 2 is of course the added true factor which makes the line of the graph shift to the left.
Figures 2-4 show the importance of investigating the factor structure for R2OLS. It shows that R2OLS cannot be interpreted without an investigation of the unexplained factor structure. Irrelevant factors in uence the R2
OLS greatly
and therefore it is no longer reliable. R2
OLS is only reliable to investigate a
no unexplained factor structure in the residuals. The following simulation high-lights the in uence of the unexplained factor structure on the R2
OLS.
This simulation uses three useless factors to estimate an observed factor model. To show the sensitivity of R2
OLS to the unexplained factor structure,
the same model is simulated but with three dierent settings of the disturbance covariance matrix V of the original factor model. First the simulation uses
V = 25 ˆV (weak factor structure), then it uses V = ˆV (factor structure)
and nally it uses V= 0.04 ˆV (strong factor structure) with ˆV the residual
covariance matrix resulting from regressing the portfolio returns on the three FF factors. The risk premia and β0sremain the same, this means that only the
covariance matrix of the disturbances varies.
Figure 5 shows the sensitivity of the distribution of R2
OLSto the unexplained
factor structure. When the unexplained factor structure in the residuals is weak the R2
OLS is very high. When the factor structure is strong the gure shows
that the R2
OLSis very low. This means that when the model has a strong factor
structure in the residuals, the R2
OLS is not a reliable statistic to measure the
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 R2 OLS Density function (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 R2 OLS Density function (b)
Figure 2: Density functions of R2
OLS and FACCHECK (the ratio of the sum of
the three largest eigenvalues of the residual covariance matrix over the sum of all eigenvalues) when the simulation uses one of the three factors (blue line), two (light blue line) and all three factors (green line).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 R2 OLS Density function (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R2 OLS Density function (b)
Figure 3: Density functions of R2
OLS and FACCHECK (the ratio of the sum
of the three largest eigenvalues of the residual covariance matrix over the sum of all eigenvalues) when the simulation uses one useless factor (blue line), two (light blue line) and three useless factors (green line).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 R2 OLS Density function (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 R2 OLS Density function (b)
Figure 4: Density functions of R2
OLS and FACCHECK (the ratio of the sum of
the three largest eigenvalues of the residual covariance matrix over the sum of all eigenvalues) when the simulation uses one valid factor (blue line), one valid and one irrelevant factor (green line), and one valid factor and two irrelevant factors (light blue line).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R2 OLS Density function
Figure 5: Density functions of R2
OLS when the simulation uses three useless
factors and there is a factor structure (light blue line), strong factor structure (green line) and a weak factor structure (blue line).
5 Tests of risk premia
To test the risk premia it is important to identify the correct test statistics and investigate if these test statistics perform correctly. According to Kleibergen (2009) the FM risk premia estimator does not have a normal limiting distri-bution in case of small and or many β0s. Test statistics that are based on the
asymptotic normality of these risk premia estimators, for example the Wald t statistic, therefore do not converge to a normal or a χ2 limiting distribution in
these cases. This means that conclusions based on those type of test statistics are unreliable in the case of small and/or many β0s.
Kleibergen (2009) suggests several alternative test statistics to test a hypoth-esis on the risk premia, for example H0: λF = λF,0,whose limiting distributions
are not eected by the value or number of β0s. The λ
1 is removed because
re-searchers are only interested in the risk premia λF. This is accomplished by
removing the return of the n-th asset and taking all other asset returns in devi-ation from the return on the n-th asset. The moment conditions become:
E(Rt) = BλF,
cov(Rt, Ft) = Bvar(Ft),
E(Ft) = µF
(5.1)
with Rt = R1,t− ιn−1Rnt and B = β1− ιn−1βn, for Rt = (R01t...R0nt)0,β =
(β1t0 ...βnt0 )0; Rt:(n-1)" 1, Rnt:1"1, β1:(n-1)"k, βn:1"k. Under H0: λF = λF,0,
a least squares estimator for B is given by ˜ B = T X t=1 Rt( ¯Ft+ λF,0) T X j=1 ¯ (Fj+ λF,0)( ¯Fj+ λF,0)0 −1 (5.2) GLS-LM statistic
The GLS risk premia estimator is: ˜ λ1 ˜ λF ! = " ιn...ˆβ 0 ˆ Ω−1 ιn...ˆβ #−1 ιn...ˆβ 0 ˆ Ω −1¯ R (5.3)
where ˆΩ is a least squares covariance matrix estimator of Ω, ˆΩ = 1 T −k−1
PT
t=1( ¯Rt−
ˆ
βFt)( ¯Rt− ˆβFt)0.This GLS risk premia estimator is invariant to transformations
of the asset returns so an invariant LM statistic is obtained by incorporating the inverse of the covariance matrix. This leads to the following test statistic:
GLS − LM (λF,0) = T 1 − λ0F,0Q(λˆ F)−1F FλF,0 ( ¯R − ˜BλF,0)0Σ˜ −1˜ B( ˜B0Σ˜ −1˜ B)−1B˜0Σ˜ −1 ( ¯R − ˜BλF,0) ∼ χ2(k) (5.4)
The GLS-LM statistic is invariant to transformations of the asset returns. The GLS-LM statistic does however depend on the inverse of the covariance matrix Σ which can be of a large dimension and therefore di cult to obtain.
Factor AR statistic
When the disturbances in a linear factor model are independently normally dis-tributed with zero mean and covariance matrix Ω, it is possible to construct the likelihood function of λ1, λF,β and Ω. Under this assumption it is also possible
to construct the likelihood function for a linear factor model in which the ex-pected returns are not restricted to be equal to ιnλ0+ βλF. After concentrating
with respect to λ1 and β, the dierence between the logarithms of the
likeli-hoods of the unrestricted linear factor model and the restricted factor model under H0: λF = λF,0 is proportional to : F AR(λF,0) = T 1 − λ0 F,0Q(λˆ F)−1F FλF,0 ( ¯R − ˜BλF,0)0Σ−1( ¯R − ˜BλF,0) ∼ χ2(n − 1) (5.5) This statistic is referred to as the factor AR statistic (FAR) since it is similar to the Anderson-Ruben statistic in the instrumental variables regression model. The FAR statistic converges to a χ2(n − 1) distributed random variable when
the sample size gets large for all values of the β0s.
Conditional likelihood ratio test statistic
The likelihood ratio statistic to test H0: λF = λF,0against H1: λF6=λF,0equals
twice the dierence between the logarithms of the concentrated likelihoods under H0 and H1:
When k=1, the likelihood ratio test statistic can be speci ed as CLR(λF,0) = 1 2[F AR(λF,0) − r(λF,0)+ + q F AR(λF,0) + r(λF,0)2− 4r(λF,0)J F AR(λF,0) ] (5.7) with r(λF,0) = ˆQ(λF)F FB˜0Σ˜ −1˜
B, which corresponds with the conditional likelihood ratio statistic of Moreira for the linear instrumental variables regres-sion model. The conditional likelihood statistic converges to a random variable with F AR(λF,0) d → ϕk+ ϕn−k−1 , JF AR(λF,0) d → ϕn−k−1, ϕk ∼ χ2(k) and ϕn−k−1∼ χ2(n − k − 1). T-test statistic
The T-test statistic tests H0 : λF = λF,0 against H1 : λF6=λF,0 similar to the
previous test. The dierence is that the test statistic is based on the convergence to a normal distribution. The test statistic is speci ed as follows:
t = ˆ λF− λF,0 q var(ˆλF) T (5.8) with var(ˆλ1, ˆλF) = " ιn...ˆβ 0 ιn...ˆβ #−1 ιn...ˆβ 0 ˆ Θ ιn...ˆβ " ιn...ˆβ 0 ιn...ˆβ #−1 , ˆ Θ = ˆΩ(1+ˆλ0F(T1PTt=1F¯tF¯t0)−1λˆFand ˆΩ = T −k−11 P T t=1( ¯Rt− ˆβ ¯Ft)( ¯Rt− ˆβ ¯Ft)0. MLE-Wald statistic
This research also investigates the MLE estimate besides the FM estimate that is described earlier. The MLE results from the eigenvectors that belong to the largest roots of the characteristic polynomial:
|θ " 1 T T X t=1 1 ¯ Ft ! 1 ¯ Ft !0# − " 1 T T X t=1 1 ¯ Ft ! R0t # Σ−1 " 1 T T X t=1 Rt 1 ¯ Ft !0# | = 0 (5.9) with Σ = 1 T PT t=1RtR 0 t: ¯ λF = W−1 0 2 w 0 1 (5.10)
and where w1 : 1 ∗ k, W2 : k ∗ k and
w1
W2
!
contains the k eigenvectors that belong to the k largest roots of the characteristic polynomial, w1
W2
! = (q1...qk), with qi, i = 1, ..., k, the (k + 1) ∗ 1 eigenvectors that belong to the k
largest (of the (k+1) roots of the characteristic polynomial. The covariance matrix of the MLE equals
var(¯λF) = 1 T 1 + ¯λ0F 1 T T X t=1 ¯ FtF¯t0 !−1 ¯ λF ˜B0Σ˜−1B˜ −1 (5.11) where ˜Σ uses the MLE for ¯λF and ˜B, and has led to errors in variables
correction of the covariance matrix of the FM estimator by Kleibergen (2006). The Wald statistic takes the following form:
F M − W (λF,0) = (ˆλF− λF,0)0var(ˆλ)−1(ˆλF− λF,0) ∼ χ2(k) (5.12)
5.1 Power and size simulation
This research analyses the power and size of the above described statistics in a simulation experiment similar to Kleibergen (2006). The asset returns for the experiment are generated from the one factor asset pricing model
Rt= ιnλ1+ β( ¯Ft+ λF) + t, t = 1, ..., T, (5.13)
with Rt the n" 1 vector of excess asset returns, β the n " 1 vector of β
0s,
λ1 the zero β return, λF the risk premium and t the disturbances which are
independently normally distributed with zero mean and covariance matrix Ω. Ωequals the least squares covariance matrix of the original one factor model. This experiment uses the data from the Kenneth French website which indicates that n = 25 and T = 421. ¯Ft is the demeaned return on the value weighted
portfolio, λ1 equals the FM estimate ˆλ1.
The linear factor model in equation 4.8 is generated using dierent values of beta. Kleibergen (2006) suggests that tests that are based on normal limiting distributions show peculiar behavior when the β0s become very small or the
number of β0sbecomes very large. Therefore the size of the β is varied for this
with ˆβ the least squares estimator for the β0sof the asset returns with respect
to the value weighted portfolio's. In the experiment λF takes a range of values
and the previously discussed test statistics are tested for H0: λF = 3with 95%
signi cance.
Figure 6 shows the results of the power curve simulations for the various values of β. The gure combine the power curves of the t-test and Wald-test, which are tests that use a normal limiting distribution, in the graphs on the left side. The power curves of the FAR, GLS-LM and CLS test statistics that do not use a normal limiting distributions are shown in the graphs on the right side. These should not be eected by a small value of β.
Figure 6a and 6b show the power curves of the test statistics for β = ˆβ. The gures show that the size of all test statistics are approximately 5% at λF = 3.
When the value of β is decreased in gures 6c and 6d the graphs show that the power curve of the t-test shifts to the right and the power curve of the Wald-test shifts upwards. Which means they become size distorted. The rejection frequency of the t-test is low at a value of λF which deviates largely from the
value for which it is being tested. The deviation becomes larger as the value of β becomes smaller. So there is a large decline in power of the t-test statistic at values of λF which are considerably dierent from the hypothesized value of
three. The graph of the Wald-test shows that the rejection frequency increases at the speci ed value of λF when the value of β is decreased. This means that
it rejects the null hypothesis more often then it should and therefore leads to false conclusions.
The gure also shows the graphs of the GLS-LM, FAR and CLR statistics. The test statistics show no power distortion which is expected as these statistics are designed to be able to cope with small β0s.
The power curves in gure six clearly show that the proposed test statistics in this section outperform the t-test and Wald-test statistics in terms of power. They do not sacri ce power to improve the size of the test statistic. It also shows that the t-test and Wald-test become power distorted when the value of β becomes very small.
0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λF Rejection frequency
(a) Power curves of T-test (blue line) and Wald-test (red line) for β = ˆβ 0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λF Rejection frequency
(b) Power curves for GLS-LM test (yellow line), FAR test (green line) and CLR test (purple line) for β = ˆ β −20 −15 −10 −5 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λF Rejection frequency
(c) Power curves of T-test (blue line) and Wald-test (red line) for β = 0.25 ˆβ −20 −15 −10 −5 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λF Rejection frequency
(d) Power curves for GLS-LM test (yellow line), FAR test (green line) and CLR test (pur-ple line) for β = 0.25 ˆβ
−30 −20 −10 0 10 20 30 40 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λF Rejection frequency
(e) Power curves of T-test (blue line) and Wald-test (red line) for β = 0.1 ˆβ −30 −20 −10 0 10 20 30 40 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λF Rejection frequency
(f) Power curves for GLS-LM test (yellow line), FAR test (green line) and CLR test (purple line) for β = 0.1β
6 Testing methods for a selection of risk premia
The tests described in section ve test hypotheses that are speci ed on all risk premia in the factor model. This means that H0 : λF = λF,0. In this research
it is necessary to test a hypothesis on a subset of the parameters instead of a hypothesis speci ed on all of the parameters. This leads to H0 : θF = θF,0,
where λF =
vF0 ...θF0
0
,with vF : kv∗ 1; θF : kθ∗ 1and k = kv+ kθ.Kleibergen
(2006) shows that the limiting distribution of the subset statistics in the linear instrumental variables regression model are bounded by limiting distributions identical to the distributions in section ve, when the left out parameters (vF)
are estimated by maximum likelihood. The maximum likelihood estimator for vF given H0 : θF = θF,0 results in a straightforward manner from the
charac-teristic polynomial of equation 5.9 when Rt and
1 ¯ Ft
!
are replaced by the residuals of the time series regression of Rt and
1 ¯ Ft ! on ¯Ft,2+ θF,0 with ¯ Ft= ¯ Ft,10 ... ¯Ft,2 0
, ¯Ft,1: kv∗ 1, ¯Ft,2: kθ∗ 1.This leads to the following revised
statistics.
Subset test statistics 1. FM-LM((˜vF(θF,0)
0...
θF,00 )0)and GLS-LM((˜vF(θF,0) 0...
θ0F,0)0)are bounded from above by χ2(k
θ)distributions.
2. FAR((˜vF(θF,0) 0 ...
θ0F,0)0)is bounded from above by a χ2(n−k
v−1)distribution. 3.CLR((˜vF(θF,0)...θ 0 F,0) 0 )given r((˜vF(θF,0)...θ 0 F,0) 0
)is bounded from above by
1 2[ϕkθ+ ϕn−k−1− r((˜vF(θF,0)...θ 0 F,0) 0 )+ r ϕkθ+ ϕn−k−1+ r((˜vF(θF,0)...θ 0 F,0) 0 )2− 4r((˜v F(θF,0)...θ 0 F,0) 0 )ϕn−k−1]
where ϕkθ and ϕn−k−1 are independent χ
2(k
θ) and χ2(n − k − 1)
dis-tributed random variables and r((˜vF(θF,0)...θ 0
F,0)
0
) is the smallest eigenvalue of ˆ Q((˜vF(θF,0)...θ 0 F,0) 1 2 0 F FB˜0Σ˜ −1˜ B ˆQ((˜vF(θF,0)...θ 0 F,0) 1 2
F F and ˜B, ˜Σare computed using
((˜vF(θF,0)...θ 0
F,0)
0 ).
FM MLE λRm−Rf -0,98 (6.7992) -1,677 (0,0957)
λSM B 0,038 (0.6773) -0,004 (0,0002)
λHM L 0,32 (0.7503) 0,819 (0,0026)
Table 3: Estimates of the risk premia for the Kenneth French data set (Standard errors between brackets).
These revised statistics show that the previously described statistics can be used to conduct tests on a selection of risk premia. This result holds for all values of the β0sdue to the immunity of the statistics to small and or many β0s.
6.1 Investigation of the risk premia for the Kenneth French
data set
When the values of λF,0 or θF,0 are varied, the previously discussed statistics
can be used to construct con dence sets for the risk premia. The (1 − α)*100% con dence sets contain all values of λF,0 or θF,0 for which the value of the
statistic that is used to test H0 : λF = λF,0 or H0 : θF = θF,0, is below its
(1 − α) ∗ 100%asymptotic critical value.
The asymptotic con dence sets, for the estimated risk premia, are con-structed for the data that is obtained from the website of Kenneth French. This research uses the factor model
Rt= ιnλ1+ β( ¯Ft+ λF) + t, t = 1, ..., T, (6.1)
to describe the return of twenty- ve size and book to equity ratio sorted portfolios with T = 421 observations. The k" 1vector of demeaned factorsF¯t contains three factors (k=3): the excess market return, SMB and HML. The analysis of the factor structure of the data shows that these three proxy factors greatly reduce the residual variance. Therefore these factors are used to estimate the accompanied risk premia.
Table 3 shows the risk premia estimates of the excess return (λRm−Rf), the SMB (λSM B) and the HML (λHM L). The risk premia are estimated using the
Fama and MacBeth two pass procedure (FM) and the MLE method. There is a dierence between the FM and the MLE estimates due to the fact that there is a downward bias when the β0sare small or many for the FM procedure. The
risk premium of the excess market return is rather large in comparison to the other risk premia.
The estimates in table three are used in combination with the discussed statistics to construct 95% asymptotic con dence sets for the dierent risk pre-mia. Figure seven shows the p-value plots of the tests on the risk premia using the dierent test statistics. Figures 7a and 7b show the p-value plots of the test statistics of the hypotheses H0: λRM −RF = λRM −RF,0,where λRM −RF is the
risk premium for the excess market return, for a range of values of λRM −RF,0.
The gures 7c and 7d show the p-value plots of H0 : λSM B = λSM B,0, where
λSM B is the risk premium for SMB for a range of values of λSM B,0.Finally the
gures 7e and 7f contain the p-value plots of the test of H0: λHM L= λHM L,0,
where λHM Lis the risk premium for HML for a range of values of λHM L,0.To
test the hypotheses we use the model described in equation 6.1 and use the sub-set test statistics to estimate one of the three risk premia. The Wald-test and t-test use the FM estimator and the CLR, FAR and GLS-LM statistics use the MLE estimator. The p-values of the statistics result from their limiting distri-bution, that is described earlier in the text. For the CLR estimater the p-values are estimated using the bounden limiting distribution given r((˜vF(θF,0)...θ
0
F,0)
0 ), which is calculated for a large amount of values of that same r((˜vF(θF,0)...θ
0
F,0)
0 ). The 95% con dence set is de ned at the intersection of the p-values with the line at 0.95. The null hypothesis is that the risk premium that is tested is equal to zero.
Figures 7a and 7b show the p-value plots of the excess market return. Fig-ure 7a shows the p-value plots for the t-test and the Wald-test. The dierence between the Wald-test and the t-test is that the t-test has a larger con dence set. Interestingly enough the Wald-test rejects the hypothesis at a 95% signif-icance level of a zero risk premium but the t-test does not. One should keep in mind that these test statistics are sensitive to small beta's and therefore not completely reliable.
Figure 7b show the p-value plots of the GLS-LM, FAR and CLR test statis-tics. We see that the con dence interval for the FAR an CLS statistics are larger than for the GLS-LM test statistic. The FAR statistic has a larger degree of freedom for the χ2 distribution then the GLS-LM statistic which causes this
eect. This leads to the GLS-LM statistic to reject the hypothesis of a zero risk premium for the excess market return. The FAR and CLR statistics do not reject this hypothesis.
The gures 7c an 7d show the p-value plots for the risk premium of SMB. For the Wald-test and t-test statistic the plots are very similar. Both show that the hypothesis of a zero risk premium is not rejected using 95% signifcance. The FAR,CLR and GLS-LM statistics are all centered around the MLE estimate. Using 95% signi cance these test statistics do not reject the null hypothesis. The simulation performed earlier show that the FAR, CLR and GLS-LM are more reliable and the conclusion therefore is that the null hypothesis is not rejected.
Figures 7e and 7f show the p-value plots for the risk premium of HML. The Wald-test 95% con dence set is slightly larger than that of the t-test. Figure 7f shows that the con dence sets of the FAR and CLR are larger than the 95% con dence set of the GLS-LM statistic. The gures show that all statistics reject the null hypothesis of a zero risk premium.
Figure 7 shows that it is important to uses size-correct statistics. Especially for the excess market return risk premium the gure shows that the test statistics may show dierent results. Although it has to be noted that for this example the size distortion problem is less of a threat than it may be using dierent factors. The factors that are used in this model have beta's that are not small enough for the size distortion to be of signi cant in uence.
−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λRF
one minus p−value
(a) One minus p-value plot for Wald-FM test and t-test testing risk premium λRM −RF −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λRF
one minus p−value
(b) One minus p-value plot for CLR, FAR and GLS-LM statistics testing risk premium
λRM −RF −0.40 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λRF
one minus p−value
(c) One minus p-value plot for Wald-FM test and t-test testing risk premium λSM B −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λRF
one minus p−value
(d) One minus p-value plot for CLR, FAR and GLS-LM statis-tics testing risk premium λSM B
−0.20 0 0.2 0.4 0.6 0.8 1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λRF
one minus p−value
(e) One minus p-value plot for Wald-FM test and t-test testing risk premium λHM L −0.20 0 0.2 0.4 0.6 0.8 1 1.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 λRF
one minus p−value
(f) One minus p-value plot for CLR, FAR and GLS-LM statis-tics testing risk premium λHM L
7 Conclusion
The simulation results of the R2
OLSfor the observed proxy factors show that this
measure is not reliable. Increasing the number of proxy factors that are incor-porated in the model leads to a higher R2
OLS. This also holds when completely
useless factors are added. A high R2
OLS indicates that the factor model is a
good t. But the unexplained factor structure in the rst pass residuals shows dierent results. Adding more useless factors does not reduce the unexplained factor structure, which means that the model does not improve. When a factor structure is absent the R2
OLS is a reliable measure. But when this is not the
case one should look carefully at the proposed FACCHECK measure.
There is a lot of literature available that suggests all sorts of observed proxy factors. I have investigated consumption growth, labor income growth, RMW and CMA. The investigation of the factor structure shows that all of these observed proxy factors do improve the R2
OLS but still leave a considerable
un-explained factor structure in the rst part residuals. This means that these variables do not contribute any additional information and are in fact useless for the model.
This research also con rms that the test statistics that are based on the FM risk premia estimator become size distorted. The simulation results of the test statistics suggested by Kleibergen (2009), which are centered around the MLE estimate, do not show the same size distortion when the β0s become small or
the number of β0sbecomes large. These tests use β0sthat are uncorrelated with
the average return vector that is used to test the risk premia. This means that these test statistics are much more reliable and may lead to other conclusion.
The tests are performed on a portfolio of twenty ve size sorted portfolio's. They show that the risk premia for the market excess return is signi cant but the risk premia of the HML and SMB are not.
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