Weak identification in linear factor models

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Faculty of Economics and Business

Weak Identification in Linear Factor Models

by

A.C.Y. Wong

Supervisor:

prof. dr. F.R. Kleibergen

Second Marker:

prof. dr. H.P. Boswijk

MSc Econometrics June 28, 2015

Abstract

Results of this thesis show that small β’s are a possible issue for linear factor models with currency portfolios. A principal component analysis on the currency portfolio returns show that the factors used in Lustig and Verdelhan (2007) leave a large unexplained factor structure in the first pass residuals from the Fama and MacBeth (1973) (FM) two pass procedure, while the factors proposed by Lustig, Roussanov, and Verdelhan (2011) do not. The second pass statistics based on the FM two pass procedure become unreliable due to the unexplained factor structure and lead to incorrect statistical inferences. Factor robust test statistics based on the maximum likelihood estimator of Gibbons (1982) are used to test hypotheses on the risk premia. These test statistics proposed by Kleibergen (2009) remain trustworthy irrespective of the values of the β’s. The 95% confidence sets show no support for the consumption-based model of Lustig and Verdelhan (2007), but do not reject it either as the 95% confidence sets consist of the whole real line. The factors proposed by Lustig et al. (2011) are also not supported nor rejected as unbounded 95% confidence sets, which consist of the whole real line are found.

Keywords: small β’s; Fama-MacBeth two pass procedure; foreign currency portfolios; factor pricing; stochastic discount factors; weak identification; principal components

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1 Introduction

Linear factor models are commonly used statistical models in finance, see Fama and MacBeth (1973), Gibbons (1982), Shanken (1992), Lustig and Verdelhan (2007), Cochrane (2009) and Lustig et al. (2011). In asset pricing, financial models are used to analyze the relationship between portfolio returns and (macro-) economic factors. These financial models imply a linear factor structure for the portfolio returns. The portfolios have varying risk exposures with respect to the factors and these varying exposures are also known as factor loadings, which are denoted by β. Linear factor models imply risk premia (factor prices) for the different factors, which are denoted by λF. The differences between expected portfolio returns are explained using the different factor loadings per portfolio, where each factor has a different risk premium.

The FM two pass procedure is used to find estimates for the risk premia (Fama & MacBeth, 1973). The first pass of the FM two pass procedure gives estimates of the factor loadings of the (macro-) economic factor by regressing the portfolio returns on the factors. In the second, pass the average portfolio returns are regressed on the estimated factor loadings from the first pass to find estimates for the risk premia.

The strength of the relationship between the expected portfolio returns and the factors can be evaluated using the ordinary least squares (OLS) R2from the second pass regression and the Wald t-statistics of the risk premia. However these second pass measures have recently been criticized as they become unreliable when the estimated β’s are small and close to zero. Kleibergen (2009) shows that the second pass Wald t-statistic becomes unreliable in the case of small β’s as it behaves in a non standard way and therefore proposes the use of robust test statistics, which remain reliable irrespective of the size of the β’s. Kleibergen and Zhan (2014) show that the R2 in the second pass can become large, but uninformative with respect to the strength of the relationship when observed proxy factors are only minorly correlated with the unobserved true factors.

In light of these findings, it is important to analyze in which areas of finance small β’s can be an issue, as second pass statistics become unreliable which lead to incorrect statistical inferences. This thesis will focus on the area of foreign currency risk premia, where such analysis has not been con-ducted yet. Lustig and Verdelhan (2007) use a consumption-based model to explain the differences between currency portfolio returns, which are sorted on interest rates. The consumption-based model implies a linear factor model, where non-durable consumption growth, durable consumption growth and market return are used as factors. Lustig et al. (2011) use an arbitrage pricing theory (APT) approach, where no economic structure is assumed contrary to consumption based models. Factors can be found using principal component analysis (PCA) and they identify two factors: average currency portfolio return and a ”slope” factor.

The goal of this thesis is to analyze to what extent the size of the β’s is an issue in Lustig and Verdelhan (2007) and how estimation results are affected. In addition, the two factors proposed by

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Lustig et al. (2011) will be constructed from the currency portfolio returns of Lustig and Verdelhan (2007) and analyzed to see to what extent these are valid factors such that use of second pass statistics is appropriate.

The thesis is organized as follows. In the second section the models used by Lustig and Verdel-han (2007) and Lustig et al. (2011) are discussed and how these models relate to linear factor models. The third section discusses application of principal component analysis to analyze the fac-tor structure of the portfolio returns. In the fourth section the effects of weak facfac-tors on the OLS R2_{are discussed. The fifth section discusses the factor robust statistics from Kleibergen (2009) and} the application of these statistics on Lustig and Verdelhan (2007). The sixth section concludes.

2 Linear Factor Models for Portfolio Returns

2.1 Consumption-based Models

In asset pricing, consumption-based models are used to explain the price of an asset which has payoffs in the future through an economic structure where an investor must decide how much to consume, save and what kind of assets to hold in his portfolio. If the investor decides to consume less today it will result in a marginal utility loss, however this enables the investor to save money and buy more of the asset. This results in a marginal utility gain as the investor can consume more of the asset’s future payoff. In equilibrium the marginal utility loss of consuming less today should equal the marginal utility gain of consuming more tomorrow through the use of the asset. If this equality does not hold, the investor should invest more or less into the asset. The price of the asset should therefore be equal to the expected discounted value of its payoff, where the marginal utility of the investor is used to discount the payoff. Risk corrections to the asset prices should depend on the covariance of the asset payoffs with consumption. Other things equal, an asset which performs badly during recessions when consumption is low and marginal utility is high is less desirable than an asset that performs badly during booms when consumption is high and marginal utility is low. During a recession an investor wishes to have an asset which performs well as his consumption is low such that an additional unit of consumption results in a large marginal utility gain. The investor is therefore considering the following maximization problem

max ζ u(ct) + Etδu(ct+1) s.t. (1) ct= et− ptζ (2) ct+1= et+1+ xt+1ζ, (3)

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time t, δ is the subjective discount factor, et is the initial consumption level and ζ is the amount of the asset the investor can buy at time t for a price of pt. The consumption one period ahead ct+1 is random, as the investor does not know his wealth one period ahead. By substituting the restrictions into the objective function, the following first order condition can be found by taking the derivative of the objective function with respect to ζ

−u0(ct)pt+ Etδu0(ct+1)xt+1 = 0, (4) which can be rewritten as

pt= Etδ

u0(ct+1) u0_(c

t)

xt+1. (5)

Equation (5) is the central asset pricing formula. It gives the market price ptof the asset given the payoff xt+1 and consumption choice ct and ct+1. By defining the stochastic discount factor (SDF) mt+1as mt+1≡ δ u0(ct+1) u0_(c t) , (6)

the basic pricing equation (5) can be further simplified to

pt= Etmt+1xt+1. (7)

Equation (7) is a generalization of standard discount factor concepts. All risk corrections can be incorporated into a single SDF. As mt+1is random it is not known with certainty at time t. The correlation between the payoff xi

t+1of the asset i and the random components of the SDF generate asset-specific risk corrections. In further derivations the time subscript is suppressed, if it is not necessary to explicitly state it.

Returns are often used in finance instead of prices of assets. The gross return of an asset is defined as

Rt+1≡ xt+1

pt

. (8)

A return can be interpreted as a payoff with a price of one, therefore returns satisfy the following equality

1 = E(mR). (9)

Instead of gross returns it is also common in finance to analyze excess returns. An investor earns an excess return Re_{by shorting one portfolio and investing the proceeds in another portfolio. This}

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is called a zero-cost portfolio such that these portfolios satisfy

0 = E(mRe). (10)

The risk-free rate is known at time t + 1, so 1 = E(mRf) = E(m)Rf. This implies that the risk-free rate is given by

Rf = 1

E(m). (11)

The SDF in equation (6) is expressed in terms of marginal utility growth, which is not directly measurable. Asset pricing is therefore concerned with finding good proxies for aggregate marginal utility growth. The proxies are also known as factors ft+1and they are used to linearly approximate the SDF such that

mt+1≡ δ

u0(ct+1) u0_(c

t)

≈ a + b0ft+1 (12)

is a reasonable and an economically interpretable approximation. In Appendix 8.1 it is shown that the linear model for the SDF m is equivalent to an expected return-beta model,

E(Ri) = γ + λ0βi⇔ m = a + b0f. (13) The expected return-beta model can be estimated with the FM two pass procedure. In the first pass, estimates of β are computed using a linear factor model. In the second pass, average portfolio returns are regressed on the estimates of β to obtain estimates for λ. The FM two pass procedure will be discussed in further detail in section 2.6.

2.2 Lustig and Verdelhan (2007)

Consider an US investor who is borrowing at home at the US interest rate and lending abroad at a foreign interest rate. If the foreign interest rate is higher than the US interest rate, then rational investors expect that the foreign currency depreciates against the dollar by the difference between the two interest rates. This is known as the uncovered interest rate parity (UIP) and due to this the US investor should earn zero excess return, however the UIP is often violated in the data. The data shows that US investors almost always earn higher excess returns in foreign currency markets, when foreign interest rates are higher than the US interest rate.

Lustig and Verdelhan (2007) use a consumption-based model to explain the violation of the UIP and show that excess returns are a compensation for the US investor as he takes on more US consumption growth risk. In the general consumption-based model the pricing equation with excess returns can be written as

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0 = E(mRet+1) = E(m)E(R e t+1) + cov(m, R e t+1) (14) or E(Rt+1e ) = − cov(m, Re t+1) E(m) (15) = −cov δu 0_(c t+1), Ret+1 E(m)u0_(c t) (16) = −cov δu 0_(c t+1), Ret+1 E δu0(ct+1) u0_(c t) u 0_(c t) (17) = −cov u 0_(c t+1), Ret+1 E u0_(c t+1) , (18)

where m has been substituted by equation (6). The utility function is monotone increasing and concave, so the marginal utility u0(c) declines as c increases. Lustig and Verdelhan (2007) show that high foreign interest rate currencies depreciate on average, when US consumption growth is low, i.e. returns of these currencies are positively correlated with consumption. Therefore the covariance between marginal utility and returns is negative. It follows from equation (18) that investors require a positive excess return for high interest foreign currencies.

Lustig and Verdelhan (2007) construct eight currency portfolios, which are sorted on interest rates. They use portfolios instead of individual currencies to diversify away the currency specific risk in order to eliminate the currency specific component which is unrelated with interest changes. The SDF is approximated with a linear model and this implies a linear factor model as discussed earlier. Non-durable consumption growth, durable consumption growth and stock market return are used as factors for their linear factor model.

2.3 Arbitrage Pricing Theory

The Arbitrage Pricing Theory (APT) starts from a statistical characterization to explain differences between asset returns (Ross, 1976). Asset returns are driven by two sources: the common com-ponents between the assets, known as factors and the idiosyncratic movement of a specific asset. Expected returns of different assets should only be related to the covariance between assets and risk factors. The idiosyncratic risks of an asset should not be priced as investors can hold well diversified portfolios such that these risks are diversified away. Investors should therefore only be compensated for the risk associated with the common risk factors. The appeal of APT over consumption-based models is that no economic structure is assumed, while pricing implications for asset returns can still be derived.

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The return generating process for portfolio returns is driven by k unobserved factors in the following form

rit= µRi+ βi1f1t+ ... + βikfkt+ εit, i = 1, ..., N, t = 1, ..., T, (19) where rit is the return of the i-th portfolio in period t, µRi is the mean return of the portfolio, fjt is the realization of the j-th risk factor in period t, βij is the factor loading of j-th factor onto the i-th portfolio, εitis the idiosyncratic risk of the i-th portfolio in period t, N is the number of portfolios and T is number of time periods. This can be written in vector notation

Rt= µR+ βFt+ εt, (20) where Rt= (r1t, ..., rN T)0, µR= (µR1, ..., µRN)0, Ft= (f1t, ..., fkt)0, εt= (ε1t, ..., εN t)0 and β =     β11 · · · β1k .. . . .. ... βN 1 · · · βN k     . (21)

Under the assumption that Ft and εtare i.i.d. with finite variance and uncorrelated for all t, the covariance matrix of the portfolio returns is

VRR= βVF Fβ0+ Vεε, (22)

where VRR, VF F and Vεε are the N × N , k × k and N × N covariance matrices of the portfolio returns, the factors and the residuals respectively. In the absence of arbitrage opportunities it is shown in Appendix 8.2 that the linear factor model implies an expected return-beta model

E(Rt) = ιNλ0+ βλF, (23)

where ιN is a N-dimensional vector with ones, λ0 is the k-dimensional zero-β return vector and λF the k-dimensional vector with factor risk premia (Ross, 1976). Conditions that the stochastic discount factor should satisfy can be derived from the expected return-beta model as these models are equivalent to linear models for stochastic discount factors.

2.4 Linear Factor Models with Observed Proxy Factors

As the k factors are unobserved it is common to use observed factors which can serve as a proxy for the unobserved factors. These observed proxy factors can be asset return based factors and (macro) economic factors. The resulting model is similar to equation (20), but there are m observed proxy factors now instead of k unobserved factors, which leads to

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Rt= µ + BGt+ Ut, (24) where µ is a N -dimensional vector with constants, B is the N × m dimensional matrix with observed proxy factor loadings, Gt = (g1t, ..., gmt)0 is a vector with observed proxy factors and Ut= (u1t, ..., uN t)0 is the vector with residuals.

2.5 Lustig et al. (2011)

Lustig et al. (2011) start from an APT approach to identify common risk factors for currency portfolios. The observed proxy factors can be found by applying principal component analysis on the covariance matrix of the currency portfolio returns. By analyzing the factor structure of the currency portfolios they find that the first two principal components explain most of the variation in the currency portfolio returns. The first principal component is identified as a level factor as all portfolios load evenly on this factor, so they use the average currency portfolio returns as the first factor. The second principal component is identified as a slope factor, because the weights from low to high interest rate currency portfolios are monotone increasing from negative to positive. They argue that this is similar to the return of a zero-cost strategy of an investor who goes long in the highest interest rate currency portfolio and short in the lowest interest rate currency portfolio. The slope factor is labeled as the carry trade risk factor and denoted with HMLF X.

2.6 FM Two Pass procedure

Fama and MacBeth (1973) propose a two pass procedure to estimate B, λ0and λF for the observed factor model, which will be discussed in this subsection.

In the first pass an estimate for B is obtained by regressing the portfolio returns Rt on the observed proxy factors Gt, the OLS estimator for B in equation (24) is:

ˆ B = T X t=1 ¯ RtG¯0t T X t=1 ¯ G0_tG¯t !−1 , (25)

where ¯Rt= Rt− ¯R, ¯R =PT_t=1Rt and ¯Gt= Gt− ¯G, ¯G =PT_t=1Gt. The OLS estimator for µ is

ˆ

µ = ¯R − ˆB ¯G. (26)

In the second pass the average return ¯R is regressed on ιN and ˆB to find estimates for λ0and λF ˆ_λ₀ ˆ λF ! = (ιN... ˆB)0(ιN... ˆB) −1 (ιN... ˆB)0R.¯ (27)

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The covariance matrix of ˆλ = (ˆλ0, ˆλF)0 can be found in the following way X ≡ (ιN... ˆB), (28) cov(ˆλ) = 1 T(X 0_X)−1_X0_cov T(Rt)X(X0X)−1, (29) with Rt= X ˆλ + et, (30)

where etis a N × 1 vector of residuals such that

covT(Rt) = 1 T T X t=1 ete0t= ˆΣ, (31) cov(ˆλ) = 1 T(X 0_X)−1_X0_ΣX(X_ˆ 0_X)−1_. ₍₃₂₎

3 Principal Component Analysis

This thesis follows the approach of Kleibergen and Zhan (2014) to analyze the factor structure in the portfolio returns. The factors simultaneously affect the portfolio returns, therefore principal component analysis can be applied to identify the number of factors. There exists a vast literature on the identification of the number of factors, see e.g. Anderson (1958), however this is beyond the scope of this thesis. Only few elements of this literature are applied to show how weak factors leave a large unexplained factor structure, which affects the R2

OLS in the FM two pass procedure. The spectral decomposition of the covariance matrix of the portfolio returns is constructed to identify the number of factors

VRR= P ΛP0, (33)

where P = (p1, ..., pN)0 is the N × N orthonormal matrix containing eigenvectors and Λ is the N × N diagonal matrix with eigenvalues. The number of eigenvalues that are distinctly larger than the others provide an estimator of the number of factors.

Furthermore principal component analysis can be used to analyze whether the observed proxy factors capture the underlying factor structure sufficiently. This can be done by regressing the portfolio returns Rton the observed proxy factors Gt, such that an estimate for Vεε can be found using

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ˆ Vεε= 1 T − 1 T X t=1 (Rt− ˆµ − ˆBGt)(Rt− ˆµ − ˆBGt)0. (34) If the observed proxy factors capture the factor structure well, then the principal components of Vεε should be small and uninformative as this implies that there is no unexplained factor structure left.

As in Kleibergen and Zhan (2014) three metrics will be used to analyze the factor structure. A measure for the presence of a factor structure with p factors is the fraction of total variation in the currency portfolio explained by the p largest principal components. The total variation is the sum of all principal components. The factor structure check is defined as

FACCHECK = λ1+ λ2+ ... + λp λ1+ λ2+ ... + λN ,

with p < N and λ1> λ2> ... > λp> ... > λN the principal components in descending order. The principal components of the covariance matrices can also be used to test for the significance of B. The likelihood ratio (LR) statistic tests whether the parameters of the observed proxy factors are equal to zero, i.e., H0: B = 0 against the alternative that they are unequal to zero H1: B 6= 0. The LR test is equal to

LR = T log(| ˆVRR|) − log(| ˆVεε|) = T N X i=1

log(λi,RR) − log(λi,εε), (35)

where ˆVRRis the estimated covariance of the portfolio returns and ˆVεε is the estimated covariance matrix of the residuals after regressing the portfolio returns on the factors G and it can be estimated using equation (34). λi,RR and λi,εε are the principal components of covariance matrices of the portfolio returns and the residuals respectively with i = 1, 2, ..., N . The large sample distribution of the LR statistic in equation (35) is χ2_{(3N ) distributed under the null hypothesis.}

The pseudo-R2 _{can be used as a goodness of fit measure to see what percentage of the total} variation of the portfolio returns is explained by the observed proxy factors and it can be computed using the the principal components of the covariance matrices of the portfolio returns and the residuals. The pseudo-R2 _{is defined as}

pseudo-R2= 1 − PN i=1λi,εε PN i=1λi,RR . (36)

3.1 Data

The data is obtained from Lustig’s personal website and comprises of currency portfolio returns, Fama-French factors and (non-) durable consumption growth data. The Fama-French factors can be

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obtained from Kenneth French’s personal website http://mba.tuck.dartmouth.edu/pages/faculty/ ken.french/data library.html. Consumption growth data is obtained from Yogo (2006) from his personal website. The average currency portfolio return factor and the HMLF X factor proposed by Lustig et al. (2011) can be constructed from the portfolio returns. The former is constructed by computing the average return for each time period and the latter by subtracting the return of the lowest interest rate currency portfolio from the highest interest rate currency portfolio.

3.2 Principal Component Analysis Results and Discussion

This subsection discusses the results found by applying principal component analysis on the data from Lustig and Verdelhan (2007). In Table 1 the principal components of the portfolio returns are given in the second column and in the third column the total variation explained by the principal component is given in percentages. The first two principal components are 292.52 and 120.84 and explain 75.06% of the total variation. The third principal component is 41.28 and only explains 7.50% of the total variation. This implies that the factor structure is driven by two factors.

i λi,Port % Explained 1 292.52 53.12 2 120.84 21.94 3 41.28 7.50 4 31.61 5.74 5 21.27 3.86 6 18.49 3.36 7 13.22 2.40 8 11.49 2.09

Table 1: Principal components of the estimated covariance matrix of the portfolio returns ˆVRR.

The portfolio returns are regressed on the three factors proposed by Lustig and Verdelhan (2007) to obtain the residuals. The residuals are used to estimate the covariance matrix Vεε, whose principal components are reported in Table 2. The three factors of Lustig and Verdelhan (2007) do not seem to capture factor structure of the portfolio returns well, as the factor structure check with the two largest principal components equals 74.82%, which indicates that a large unexplained factor structure remains.

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i λi,Res % Explained 1 273.22 52.34 2 117.32 22.48 3 40.43 7.75 4 30.40 5.82 5 20.15 3.86 6 16.28 3.12 7 13.13 2.52 8 11.05 2.12

Table 2: Principal components of the estimated covariance matrix of the residuals ˆVεε.

Table 3 reports four different LR statistics and a pseudo-R2_{. The first LR statistic tests the} significance of all parameters in B of the three factors of Lustig and Verdelhan (2007) and from the table it follows that the LR statistic is highly significant. The other three LR statistics test the significance of the factors separately. To test the significance of a specific factor the principal components of the covariance matrix of the portfolio returns are replaced with the principal com-ponents of the covariance matrix of the residuals that result from regressing the portfolio returns on the two other factors. The LR statistics, where each factor is tested separately are all highly significant. The size of the reported LR statistics is not large, when taken into account that multiple parameters are tested simultaneously for significance. This is another indication that the factor structure is not captured well by the three factors of Lustig and Verdelhan (2007) and this is also supported by the pseudo-R2 _{as only 5.22% of the variation is explained by the three factors.}

LR against raw 1436.5

(0.0000) LR against durables and market 250.3

(0.0000) LR against non-durables and market 898.6

(0.0000) LR against non-durables and durables 305.6

(0.0000)

pseudo-R2 _0.0522

Table 3: LR statistic for testing H0: B = 0 vs. H1: B 6= 0 with p-values reported between brackets for all parameters, each factor separately and pseudo-R2_{of the full model.}

The estimates of the β’s and the factor prices λF can be found in Table 4 and the estimates of the factor prices are highly significant, except for the market factor which is line with the results found by Lustig and Verdelhan (2007). The second pass R2

OLS of 0.8763 is also reported in Table 4, which indicates that their exists a strong relationship between the factors and portfolio returns. The results stipulate the problem with second stage statistics when weak factors are used. From

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the results of the previous three tables it is clear that the factors of Lustig and Verdelhan (2007) do not capture the factor structure well, however significant results are found when statistical inference based on the second stage statistics is done.

i 1 2 3 4 5 6 7 8 Factor Prices Non-durable 0.0501 0.6658 -0.7294 -0.7816 0.0889 -0.8649 -0.1297 -1.4327 2.2642 (0.8776) (0.9108) (0.8985) (1.1303) (1.0263) (1.1086) (1.0687) (1.7024) (0.8079) Durable 0.1114 0.1322 1.0571 1.1263 0.5735 1.1952 1.3613 1.5664 4.5747 (0.6420) (0.6662) (0.6573) (0.8268) (0.7508) (0.8109) (0.7817) (1.2453) (0.9925) Market -0.0471 -0.0241 0.0398 -0.0540 0.0285 0.0479 0.0018 0.1071 2.6350 (0.5590) (0.0580) (0.0572) (0.0720) (0.0654) (0.0706) (0.0681) (0.1084) (8.6228) R2 OLS 0.8763

Table 4: Estimated β’s of observed proxy factors and factor prices with standard errors between brackets.

In Table 5 the eigenvectors of the two largest characteristic roots are reported and they corre-spond with the findings of Lustig et al. (2011) to some extent. They identified two common risk factors by applying principal component analysis. They link the first eigenvector to a level factor as all portfolios load evenly on the first principal component and it can be interpreted as the average excess return. This is partially supported by the second column of Table 5 as large differences be-tween the loadings of some portfolios exist. For example, the difference bebe-tween the loading of the second portfolio and the last portfolio is 0.2658 in absolute value. The second principal component is a slope factor with monotone increasing weights from low to high interest rates. This is identified by Lustig et al. (2011) as the carry trade risk factor HMLF X, however from third column in Table 5 it follows that the loadings of the second eigenvector are not monotone increasing.

i p1 p2 1 0.2587 -0.1540 2 0.2027 -0.1612 3 0.3267 -0.0908 4 0.4197 -0.2326 5 0.3154 -0.1516 6 0.3801 -0.1598 7 0.3817 -0.2675 8 0.4685 0.8763

Table 5: Eigenvectors (p1 and p2) of the two largest principal components of the estimated covari-ance matrix of the portfolio returns ˆVRR.

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Lustig and Verdelhan (2007) Lustig et al. (2011)

factors factors

F -stat two largest 0.78 4607.55

principal components (0.5893) (0.0000)

Table 6: The F -statistics (with p-values between brackets) result from testing the significance of the indicated factors in the first row in a regression of the two largest principal components on them.

The F -statistics in Table 6 result from regressing the two largest principal components on the factors from either Lustig and Verdelhan (2007) or from Lustig et al. (2011). The F -statistics then result from testing H0: γ = 0 in the linear model:

Yt= c + γXt+ Vt, (37)

with Yta 2 × 1 vector that contains the two largest principal components, which is constructed by multiplying the portfolio returns R with the two eigenvectors, i.e. R (p1 ... p2) and Xtis a 3×1 vector containing the factors from Lustig and Verdelhan (2007) or a 2 × 1 vector containing the factors from Lustig et al. (2011). The (insignificant) small F -statistic in the second column reiterates that the factors from Lustig and Verdelhan (2007) do not capture the factor structure well, while the (highly significant) large F -statistic in the third column indicates that the factors from Lustig et al. (2011) do.

The principal component analysis is repeated to see whether the average currency portfolio return factor and the carry trade risk factor do capture the factor structure sufficiently. From Table 7 it can be seen that the first principal component of the estimated covariance matrix of the residuals only explains 28.49% of the variation instead of 52.34% as reported in Table 1. The total variation explained by the first two principal components drops from 75.06% to 54.21%. The large difference between the second and the third principal component is not found in Table 7, which indicates that the factor structure of the portfolio returns is captured sufficiently by the two factors of Lustig et al. (2011).

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i λi,Res % Explained 1 42.83 28.49 2 38.67 25.72 3 21.75 14.47 4 20.82 13.85 5 14.54 9.67 6 11.76 7.82 7 0.00 0.00 8 0.00 0.00

Table 7: Principal components of the covariance matrix of the residuals using the average portfolio return and HMLF X as factors.

The LR test is computed for both factors and each factor separately. The results can be found in Table 8 and it follows from the table that the two factors are highly significant, however the HMLF X factor is less significant than the average return factor. The size of the LR statistics is also much larger than those of Table 3 and also a pseudo-R2of 0.7270 is found. The results from Table 7 and 8 indicate that the two factors proposed by Lustig et al. (2011) capture the factor structure much better than the three factors proposed by Lustig and Verdelhan (2007).

LR against raw 20017.8

(0.0000) LR against HMLF X 13850.0 (0.0000) LR against average return 5495.3

(0.0000)

pseudo-R2 _0.7270

Table 8: LR statistic for testing H0: B = 0 vs. H1: B 6= 0 with p-values reported between brackets and the pseudo-R2 of the full model using Lustig et al. (2011) factors.

4 Second Pass R

2_OLS

The R2

OLS is a commonly used statistic to determine how well the variation is explained by the estimated model. The R_OLS2 of the second pass of the FM two pass procedure equals the explained sum of squares over the total sum of squares when a constant is included, which leads to the following expression R2_OLS= ¯ R0P_M ιNBˆ ¯ R ¯ R0_M ιNR¯ = ¯ R0MιNB( ˆˆ B 0_M ιNB)ˆ −1_Bˆ0_M ιNR¯ ¯ R0_M ιNR¯ , (38)

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N × N identity matrix.

Kleibergen and Zhan (2014) show that under weak factors the R2_OLS is not consistent and that it converges to a random variable. They show that large values of the R2

OLS do not necessarily have to stem from strong factors, but can also be caused by the estimation error in ˆB. The R2

OLS becomes unreliable in this case and it can lead to incorrect statistical inferences.

4.1 Monte Carlo Simulation Experiment

To analyze the behaviour of the R2

OLS in Lustig and Verdelhan (2007) a Monte Carlo simulation is conducted to estimate the density of the R2

OLS under weak factors. Estimates for β, λ0 and λF are obtained by using the average return and HMLF X as factors in the FM two pass procedure. Currency portfolio returns are generated from the factor model from equation (20) with µ = ιNλ0+ βλF by using these estimates. The disturbances εtare generated from an i.i.d. normal distribution with mean zero and covariance matrix ˆVεε, where ˆVεεis the covariance matrix of the residuals after regressing the currency portfolio returns on the two factors. The total number of Monte Carlo simulations is 10000 and for each repetition a 50 × 8 matrix with currency portfolio returns is generated to match Lustig and Verdelhan (2007).

The generated portfolio returns are regressed on the three factors of Lustig and Verdelhan (2007) to find ˆB in order to compute R2

OLSfor each Monte Carlo repetition. The density function of ROLS2 can then be estimated by using a kernel density estimator bounded between zero and one. In addition the density of FACCHECK is also estimated, which was defined earlier as the ratio of the sum of the two largest principal components over the total sum of the principal components of the covariance matrix of the residuals.

4.2 Results of Experiment

In Figure 1 the density functions for R2

OLSand FACCHECK are plotted after regressing the average return and HMLF X factors on the currency portfolio returns. The dashed line corresponds with the density function when only one factor is used and the solid line is the density function when both factors are used. As expected, when a true factor is added it follows from the right graph in Figure 1 that the density of FACCHECK shifts to the left as the remaining unexplained factor structure decreases. The left graph in Figure 1 shows that the distribution of R2

OLS shifts to the right and that it is close to one.

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Figure 1: Density function of R2

OLS and FACCHECK (the ratio of the sum of the two largest principal components of the residual covariance matrix over the sum of all principal components) when one factor (dashed) or two factors (solid) are used of Lustig et al. (2011).

In Figure 2 the densities of R2

OLS and FACCHECK are plotted for weak factors, where the dashed line corresponds with one weak factor, the dash dotted line with two weak factors and the solid line corresponds with three weak factors. From the left graph in Figure 2 it follows that the density of R2

OLS is shifting to the right, when factors are added. The R 2

OLS is high, when all three factors of Lustig and Verdelhan (2007) are used. The right graph of Figure 2 shows the density of FACCHECK and from the graph it follows that the density shifts only a little bit to the left when factors are added. The graphs in Figure 2 show that adding weak factors shifts the density R_OLS2 to the right leading to a higher R2

OLS, but this stems from the estimation error in ˆB and not from adding relevant factors to the regression as the density FACCHECK remains unchanged. Adding weak factors has little to no effect on the unexplained factor structure in the first pass residuals.

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OLS and FACCHECK (the ratio of the sum of the two largest principal components of the residual covariance matrix over the sum of all principal components) when one weak factor (dashed), two weak factors (dash dotted) or three weak factors (solid) are used of Lustig and Verdelhan (2007).

In Figure 3 a valid factor from Lustig et al. (2011) is combined with irrelevant factors from Lustig and Verdelhan (2007). Compared to Figure 2 the shift of the density of R2_OLS to the right is less pronounced, but the density of FACCHECK remains unaffected when irrelevant factors are added such that the unexplained factor structure remains.

OLS and FACCHECK (the ratio of the sum of the two largest principal components of the residual covariance matrix over the sum of all principal components) when one valid factor (dashed), one valid factor and one irrelevant factor (dash dotted) or one valid factor and two irrelevant factors (solid) are used from Lustig and Verdelhan (2007) and Lustig et al. (2011).

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using the three weak factors of Lustig and Verdelhan (2007). The strength of the unexplained factor structure can be adjusted by varying the covariance matrix Vεεof the residuals in the original model. Three settings are used: Vεε = 0.04 ˆVεε (strong factor structure), Vεε = ˆVεε (factor structure) and Vεε = 25 ˆVεε (weak factor structure). The specification of the risk premia and the estimates of the β’s remain the same and no other adjustments are made, except for the covariance matrix of the residuals. In Figure 4 the densities of R2

OLS and FACCHECK are plotted under the various settings of Vεε and it further emphasizes the sensitivity of R2OLS to the unexplained factor structure. The density of R2

OLS is close to one, when the unexplained factor structure is strong (dashed line). As the strength of the factor structure decreases the densities of R2

OLS and FACCHECK shift to the left. The R2

OLSis therefore uninformative about the strength of the relationship between the factors and the portfolio returns for models with a large unexplained factor structure in the residuals.

OLS and FACCHECK (the ratio of the sum of the two largest principal components of the residual covariance matrix over the sum of all principal components) when three weak factors are used in combination with a strong factor structure (dashed), a factor structure (dash dotted) and a weak factor structure (solid).

5 Factor Robust Test Statististics

The linear regression estimates of β of the FM two pass procedure are sensitive to the collinear-ity of the explanatory variables (factors), which implies that the risk premia are sensitive to the collinearity of the β0s. Collinearity occurs when β’s are small in absolute value and/or when the matrix of β’s is almost of reduced rank. When weak factors are used it results in small β’s such that the large sample distribution of the FM risk premia estimator will converge to a random vari-able instead of the true value. The spurious estimates are also affected by the number of currency portfolios, even when β’s are large in absolute value. If the number of currency portfolios is large it

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can result in misleading risk premia estimates. Wald tests based on the FM two pass estimator can lead to incorrect statistical inferences as the limiting distribution of the t-statistic is non normal (Kleibergen, 2009).

Kleibergen (2009) therefore proposes the use of alternative statistics whose limiting distributions are unaffected by the size of the β’s, which can then be reliably used for statistical inference. These statistics are centered around the maximum likelihood estimator (MLE) of Gibbons (1982) and they are based on the Generalized Method of Moments (GMM) and instrumental variable statistics of Anderson and Rubin (1949), Kleibergen (2002), Moreira (2003) and Kleibergen (2005). This thesis will focus on the generalized least squares - Lagrange multiplier test (GLS-LM), the factor Anderson Rubin test (FAR) and the conditional likelihood ratio test (CLR).

These proposed statistics are used for testing hypothesis on the risk premia, such as H0: λF = λF,0. The model is adjusted by deleting λ0 from it, as the focus lies on λF. This can be achieved by removing portfolio i and by taking all other returns in deviation from the returns of portfolio i, where the choice of i is arbitrary. If i = N , then this implies the following moment conditions

E(Rt) =BλF, (39)

cov(Rt, Ft) =BVF F, (40)

E(Ft) = µF, (41)

withRt= R1t− ιN −1RN t,B = β1− ιN −1β0N, Rt= (R1t0 , RN t)0, β = (β10, βN)0. R1tis a (N − 1) × 1 vector, RN tis a scalar, β1is a (N − 1) × k matrix and βN is a k × 1 vector. Under H0: λF = λF,0, the least squares estimator ˆB is defined as

ˆ B = T X t=1 Rt( ¯Ft+ λF,0)0 T X j=1 ( ¯Fj+ λF,0)( ¯Fj+ λF,0)0 −1 , (42) with ¯Ft= Ft− ¯F and ¯F =P T t=1Ft.

5.1 GLS-LM Statistic

The GLS-LM statistic is one of the factor robust test statistic proposed by Kleibergen (2009) and under H0: λF = λF,0

GLS-LM(λF,0) =

T

1 − λ0_F,0Q(λˆ F,0)−1λF,0

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where ˆQ(λF,0) = _T1 PT_t=1( ¯Ft−λF,0)( ¯Ft−λF,0)0and ˆΣ =_{T −k}1 PT_t=1( ¯Rt− ˆB( ¯Ft−λF,0))( ¯Rt− ˆB( ¯Ft− λF,0))0. The GLS-LM statistic converges to a χ2(k) random variable as the sample size increases irrespective of the values of the β’s. A possible complication while computing the GLS-LM statistic can arise from the computation of the inverse of ˆΣ, which can be of a large dimension as it depends on N . It can be shown that the GLS-LM statistic is the quadratic form of the derivative of the FAR statistic discussed below (Kleibergen, 2009). This implies that the statistic will be equal to zero at points of λF,0, which set the derivative of the FAR statistic equal to zero such as inflexion points, local minima and maxima.

5.2 FAR Statistic

The likelihood function can be constructed under the additional assumption that the errors of linear factor model are normally distributed and it can be concentrated with respect to β and λ0. Under H0: λF = λF,0 the difference between the logarithms of the unrestricted and the restricted concentrated likelihood is proportional to the FAR statistic, which is defined as

FAR(λF,0) =

T

1 − λ0_F,0Q(λˆ F,0)−1λF,0

( ¯R − ˆBλF,0)0Σˆ−1( ¯R − ˆBλF,0). (44)

The FAR statistic converges to a χ2_{(N − 1) distribution as the sample size increases. The} mini-mization of the FAR statistic is the same as the maximini-mization of the likelihood function due to the relationship between these two. Hence the MLE can be found by minimizing the FAR statistic over λF,0.

5.3 JGLS Statistic

As discussed earlier the GLS-LM statistic is zero, whenever the derivative of the FAR statistic is zero, this difficulty can be overcome with the use a pre-test to see whether the moment conditions

E(R) = BλF at λF = λF,0 hold. The JGLS statistic can be used for this and is defined as

JGLS(λF,0) = FAR(λF,0) − GLS-LM(λF,0), (45) where under H0 : λF = λF,0 the statistic converges to a χ2(N − k − 1) distribution as the sample size increases. The limiting distributions of JGLS and GLS-LM are independent of each other for all possible values of the β’s. The JGLS test can be used in conjunction with the GLS-LM test by first testing whether the moment conditions hold under λF = λF,0 at a significance level of αJGLS. If the moment conditions are not rejected, the GLS-LM can be used with a significance level of αGLS-LM. The significance level of αJGLS and αGLS-LM are chosen such that α = αJGLS+ αGLS-LM.

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5.4 CLR Statistic

The likelihood ratio test statistic is equal to two times the difference between the logarithms of unrestricted and restricted concentrated likelihoods and it can be shown that the CLR can specified as CLR(λF,0) = 1 2 FAR(λF,0) − r(λF,0) + q (FAR(λF,0) + r(λF,0))2− 4r(λF,0)JGLS(λF,0) , (46)

where r(λF,0) is the smallest eigenvalue of ˆQ(λF,0)1/20Bˆ0Σˆ−1B ˆˆQ(λF,0)1/2. As the sample size grows larger the CLR statistic converges to

1 2 ϕk+ ϕN −k−1− r(λF,0) + q ϕk+ ϕN −k−1+ r(λF,0) 2 − 4r(λF,0)ϕN −k−1 , (47) where ϕk and ϕN −k−1 are χ2(k) and χ2(N − k − 1) distributed random variables respectively. A Monte Carlo simulation is used to find the critical regions of the CLR. First, a grid of values of r(λF,0) is set up. Second, for every value of the grid the distribution of CLR is simulated by independently drawing values from a χ2_{(k) and a χ}2_{(N − k − 1) distribution. The observed value} of r(λF,0) is then compared to its specific critical value based on the empirical distribution. In Appendix 8.3 it is shown that the CLR statistic attains it minimal of zero at the MLE.

5.5 MLE

Kleibergen (2009) shows that the MLE can be found under the assumption of normally distributed errors with a fixed covariance matrix Vεε. The MLE of λF, ˜λF, is constructed from the eigenvectors of the k largest eigenvalues of the following characteristic polynomial:

θ 1 T T X t=1 1 ¯ Ft ! 1 ¯ Ft !0 − T X t=1 1 ¯ Ft ! R0 t Σ−1 T X t=1 Rt 1 ¯ Ft !0 = 0, (48) where Σ =_T1PT

t=1RtRt0. The MLE ˜λF is then defined as

e

λF = W2−1w 0

1, (49)

where w1 is a 1 × k row vector and W2 is a k × k matrix such that (q1, q2, ..., qk) = w1 W2

! with qi, i = 1, 2, ..., k the k + 1 × 1 eigenvectors that correspond with the k largest eigenvalues (k + 1 eigenvalues in total) of equation (48). The covariance matrix of the MLE is equal to

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V f λF = 1 T 1 + fλF 0_XT t=1 ¯ FtF¯t0 −1 f λF ( ˆB0Σˆ−1B)ˆ −1, (50)

where ˆΣ and ˆB are computed using the MLE fλF.

5.6 Tests on Subsets of the Factor Prices

The test statistics discussed earlier test hypotheses where all risk premia are specified (H0 : λF = λF,0 ) instead of only a subset. Hypotheses where only a subset of the parameters is specified, for example, H0: θF = θF,0, where λF = (νF0 , θF0 )0 with νF a kν× 1 vector; θF,0, θF are kθ× 1 vectors and k = kν+ kθ can be tested by substituting λF,0 with (˜νF(θF,0)0, θF,00 )0 in the test statistics. The unspecified parameters in νF are estimated with the restricted MLE ˜νF(θF,0).

5.6.1 Testing Procedure

The factor robust test statistics can be adjusted in the following way to test hypotheses where only a subset of the risk premia is specified. First, the MLE of νF, ˜νF(θF,0), can be found by replacing in equation (48)Rtand (1, ¯Ft0)0 with the residuals of the regression ofRtand (1, ¯Ft,ν0 )0 on ¯Ft,θ+ θF,0 with ¯Ft= ( ¯Ft,ν0 , ¯Ft,θ0 )

0_{, ¯}_F

t,ν is a kν× 1 vector and ¯Ft,θ is a kθ× 1. Second, λF,0 should be replaced with (˜νF(θF,0)0, θF,00 )0 in the test statistics (GLS-LM, FAR and CLR). It can then be shown that the limiting distributions of the test statistics are upper bounds for these adjusted test statistics:

1. GLS-LM((˜νF(θF,0)0, θ0F,0)

0_{) is bounded from above by a χ}2_(k

θ) distribution. 2. FAR((˜νF(θF,0)0, θ0F,0)0) is bounded from above by a χ

2_{(N − k}

ν− 1) distribution.

3. JGLS((˜νF(θF,0)0, θ0F,0)0) is bounded from above by a χ2(N − k − 1) distribution which is independent from the limiting distribution of GLS-LM((˜νF(θF,0)0, θ0F,0)0).

4. CLR((˜νF(θF,0)0, θ0F,0)0) given r((˜νF(θF,0)0, θF,00 )0) is bounded from above by

1 2 ϕkθ+ ϕN −k−1− r((˜νF(θF,0) 0_{, θ}0 F,0) 0₎ + q ϕkθ+ ϕN −k−1+ r((˜νF(θF,0) 0_{, θ}0 F,0)0) 2 − 4r((˜νF(θF,0)0, θF,00 )0)ϕN −k−1 , (51)

where ϕkθ and ϕN −k−1are independent χ

2_(k

θ) and χ2(N −k−1) distributed random variables and r((˜νF(θF,0)0, θ0F,0)0) is equal to the smallest eigenvalue of the following matrix

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T ˆQ((˜νF(θF,0)0, θF,00 )0)

1/20_B_ˆ0_Σ_ˆ−1_{B ˆ}_ˆ_Q((˜_ν

F(θF,0)0, θF,00 )0) 1/2

, (52)

where ˆB and ˆΣ are computed using (˜νF(θF,0)0, θ0F,0)0.

5.7 Confidence Sets

The factor robust test statistics can be used to construct confidence sets for the risk premia by specifying a range of values for λF,0 or θF,0. H0 : λF = λF,0 or H0∗ : θF = θF,0 is tested for each value in the range and the GLS-LM, FAR, CLR and their corresponding p-values are computed. The confidence set corresponds with the values in the range for which the p-values are larger than the significance level α. These confidence sets of the robust test statistics can have different shapes: 1. convex (λlower F , λ upper F ), 2. unbounded: (−∞, ∞), (−∞, λ lower F ), (λ upper F , ∞) and (−∞, λ lower F ) ∪ (λupper_F , ∞) and 3. empty. An empty confidence set is only possible for FAR as GLS-LM and CLR are zero at the MLE. If a convex set is found it implies that the factors are strong and that there is no evidence for small β’s. Unbounded confidence sets are an indication of weak factors, while an empty confidence set is an indication that the factors used are not exogeneous.

5.8 Factor Robust Test Statistics Results and Discussion

This subsection discusses the results found by applying the factor robust tests on the data of Lustig and Verdelhan (2007) to see whether the significant results found there using FM second pass test statistics still hold. In addition the proposed factors in Lustig et al. (2011) are also tested to see whether these are strong factors.

In Table 9 the FM two pass procedure estimates can be found in the second column, while in the third column the MLE can be found. The standard errors are reported in brackets and it should be noted that the standard errors are not corrected for the fact that the β’s are estimated, which is in line with Lustig and Verdelhan (2007) and Lustig et al. (2011). It follows from the table that there are substantial differences between the FM estimates and the MLE for the non-durable consumption growth, the non-durable consumption growth and the market factor, although the sign remains the same for the factors of interest (non-durable and durable consumption growth). Standard errors under the MLE are much larger than the FM estimates, for example the standard error for durable consumption growth is up to 3.5 times larger for the MLE compared to the FM estimator. The market factor is estimated very inaccurately under both estimation techniques as the standard errors are very large, especially for the MLE. The proposed new factors (average portfolio returns and HMLF X) also display large differences between the FM estimator and the MLE. In particular, the average return factor is very different between the two estimators as the MLE is more than twice as large as the FM estimator. In addition, the standard error for the

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average return is much larger under the MLE than the FM estimator. For HMLF X the difference is less severe and it surprising to see that the standard error is a little bit smaller under the MLE than under the FM estimator.

Factor prices (ˆλ) FM MLE

Non-durable 2.2642 3.2257 (0.8079) (2.5272) Durable 4.5747 5.5180 (0.9925) (3.6408) Market 2.6350 -6.6679 (8.6228) (20.6727) R2 OLS 0.8763 Average 4.9634 11.8249 (1.7752) (6.1167) HMLF X 3.1968 4.3669 (1.8265) (1.4205) R2 OLS 0.5264 Geometric HMLF X: Average 3.6521 8.9632 (1.9576) (4.7601) HMLF X 3.0416 4.5951 (1.2825) (1.2271) R2 OLS 0.6175

Table 9: Estimates of factor prices using the FM two pass procedure and the MLE with standard errors between brackets.

In Figure 5 the one minus p plots can be found for the Wald t-statistic, the GLS-LM statistic, the FAR statistic and the CLR statistic of the non-durable consumption growth factor, the durable consumption growth factor and the market factor. From the first two subfloats, it follows that the Wald t-statistic confidence sets both exclude zero. The confidence sets of the factor robust test statistics are unbounded, as for all statistics the curves lie beneath the 95% confidence level. The confidence sets neither support nor reject the consumption-based model of Lustig and Verdelhan (2007), since the 95% confidence sets consist of the whole real line. The GLS-LM and CLR statistic are zero at the MLE, which corresponds with the minimum of the FAR statistic.

Confidence sets are also constructed for the average return and HMLF Xfactors, but the HMLF X factor causes problems for the MLE, because of its construction. It is simply the return of the currency portfolio with the highest interest rate minus the return of the currency portfolio with the lowest interest rate. Estimates for the restricted MLE are difficult to find as numerical techniques compute numbers that are close to singularity. To resolve this issue the return of the lowest interest rate currency portfolio is subtracted from the geometric mean of the two highest interest rate currency portfolios (p(1 + Rt,N :N)(1 + Rt,N −1:N) − 1 − Rt,1:N). The FM and MLE estimates of

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the factor prices can be found at the bottom part of Table 9. In Table 9 the estimates for HMLF X hardly change, but the estimated risk premium for the average return decreases for both the FM estimator and MLE.

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Figure 5: 1 − p plot for GLS-LM (solid), FAR (dashed dotted) and CLR (dashed) statistics for various factor prices.

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Figure 6: 1−p plot for GLS-LM (solid), FAR (dashed dotted), CLR (dashed) and JGLS (diamonds) for the average return factor price.

In Figure 6 the one minus p plot is given for the average portfolio return factor. The Wald t-statistic reports a confidence set which excludes zero. The GLS-LM t-statistic is used in conjunction with the JGLS statistic. The significance level used for JGLS is αJGLS = 0.01, while for the GLS-LM statistic a significance level of αGLS-LM = 0.04 is used. For negative values of λF,0 the JGLS test is rejected and the cutoff point is around λavg ≈ 2. For values larger than the cutoff point the JGLS does not reject the moment conditions such that the GLS-LM test can be used and an unbounded confidence set is found. The unbounded confidence set of GLS-LM excludes zero, which is also the case for FAR and CLR. This implies that statistical evidence is found to support the average return factor as a significant factor, although the 95% confidence sets are unbounded for all three factor robust test statistics.

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Figure 7: 1 − p plot for GLS-LM (solid), FAR (dashed dotted) and CLR (dashed) statistics for the average return and the HMLFXfactor prices using a geometric mean for HMLFX.

In Figure 7 the one minus p plots are given for the average return and the geometric HMLFX factor. The average return graph differs considerably from Figure 6 as the confidence set of the Wald t-statistic does not exclude zero from its confidence set. Furthermore the confidence set of the factor robust test statistics are unbounded and do not exclude zero as well. This implies that in this case there is no support for the average return as a significant factor. The HMLFXfactor does show a Wald t-statistic confidence set, which excludes zero. The factor robust test statistics show unbounded confidence sets, which consist of the whole real line. Under geometric HMLFX factor no support is found for the factors of Lustig et al. (2011). Note the shape of the curve of the GLS-LM test statistic as it is zero, whenever the FAR statistic attains a minimum or a maximum. The GLS-LM is a quadratic form of the derivative of the FAR statistic, which will have a zero derivative at a maximum or minimum.

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t-statistic, GLS-LM, FAR and CLR for the factors of Lustig and Verdelhan (2007) and Lustig et al. (2011) and it summarizes the results found from the one minus p plots from Figures 5, 6 and 7. Unbounded confidence sets are found for the three factors of Lustig and Verdelhan (2007), which implies that the factors are weak. The average return factor from Lustig et al. (2011) is significant at the 95% significance level, but the confidence set is unbounded. Under geometric HMLF X, unbounded confidence sets are found for both the average return factor and the HMLF X factor.

Factor prices (ˆ

λ)

t-stat

GLS-LM

FAR

CLR

Non-durable

(0.6747, 3.8554)

(−∞, ∞)

Durable

(2.6426, 6.5301)

(−∞, ∞)

Market

(-14.3373, 26.8675)

(−∞, ∞)

Average

(1.5422, 8.4096)

(3.6428, ∞)

(4.2071, ∞)

(4.4892, ∞)

Geometric HML

F X

:

Average

(-0.1616, 7.4747)

(−∞, ∞)

(−0.4496, ∞)

(−∞, ∞)

HML

F X

(0.5301, 5.5341)

(−∞, ∞)

Table 10: 95% Confidende Interval upper and lower bounds for various test statistics. For the GLS-LM test a confidence is used of 96% with a confidence of 99% for the JGLS statistic.

6 Conclusion

The aim of this thesis is to analyze to what extent the size of the β’s is an issue in Lustig and Verdelhan (2007) and how estimation results are affected. The two factors proposed by Lustig et al. (2011) were constructed from the currency portfolio returns of Lustig and Verdelhan (2007) to see to what extent these are valid factors such that use of second pass statistics is appropriate.

A principal component analysis is applied on the currency portfolio returns to analyze the factor structure. The factors from Lustig and Verdelhan (2007) are considered to be weak. They do not capture the factor structure sufficiently, as a large unexplained factor structure remains in the first pass residuals. The factors from Lustig et al. (2011) capture the factor structure much better, as no unexplained factor structure is found in the first pass residuals.

The inadequacy of the R2

OLS with weak factors is shown with a Monte Carlo simulation. The results show that factor structure in the first pass residuals can lead to high values of R2

OLS, especially when the factor structure is very strong. The simulation shows that the addition of weak factors will shift the density of R2

OLSto the right, but that the density of the factor structure check remains unchanged. When valid factors are used the density of the factor structure check will shift to the left, which implies that the unexplained factor structure decreases.

Factor robust test statistics are used to test hypotheses on the risk premia and are compared with the FM second pass t-statistic. These factor robust test statistics remain valid irrespective of the

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values of the β’s and inference is centered around the MLE of Gibbons (1982). The consumption-based pricing model of Lustig and Verdelhan (2007) is not supported by the factor robust test statistics as all 95% confidence sets of the factors are unbounded nor is it rejected as the 95% confidence sets consist of the whole real line. The factors of Lustig et al. (2011) are also not supported nor rejected as the 95% confidence sets of the risk premia of these factors are unbounded and also consist of the whole real line, although this is based on the geometric HMLF X.

The results from the thesis show that small β’s are an issue for foreign currency portfolios and that statistical inference based on FM second pass statistics should be done judiciously. If the FM second pass statistics are used, then the factor structure of the portfolio returns should be analyzed for linear factor models with foreign currency portfolios. No unexplained factor structure should remain in the first pass residuals, if the factors are valid. However this thesis suggests the use of factor robust test statistics by Kleibergen (2009) as these remain valid for all values of the β’s such that no principal component analysis is necessary to determine the appropriateness of using FM second pass statistics.

For further research this thesis suggests the analysis of different assets (e.g. real estate returns), where (macro-) economic factors are used in a linear factor model.

7 Acknowledgement

I thank my thesis supervisor prof. dr. Frank Kleibergen for providing me with helpful comments and suggestions.

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8 Appendix

8.1 Equivalence between Expected-Return Beta Models and Linear

Mod-els for Stochastic Discount Factors

Consider the following linear model for the SDF m

m = a + b0f, (53)

with E(f ) = 0 and hence E(m) = a. Excess returns satisfy 0 = E(mRe), therefore the mean of m is not identified, such that a can be normalized to 1. This implies E(m) = 1 or m = 1 + b0 f − E(f ). The pricing equation can be rewritten in the following way to show that the linear model for the SDF is equivalent to an expected return-beta model

0 = E(mRe) (54) = E(Re) + cov(Re, f0)b (55) ⇔ E(Re_{) = −cov(R}e_{, f}0_)b ₍₅₆₎ = (cov(Re, f0)var(f )−1) | {z } β0 (−var(f )b) | {z } λ (57) = β0λ. (58)

When gross returns are considered instead of excess returns a is not normalized to 1, so E(m) = a and the following pricing equation is satisfied

1 = E(mR). (59)

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1 = E(mR) (60) = E(m)E(R) + cov(m, R) (61) ⇔ (62) E(R) = 1 E(m)− cov(m, R) E(m) (63) = 1 a− cov(a + b0f, R) a (64) = 1 a− cov(b0f, R) a (65) = 1 a− E(Rf0)b a , (66)

where the final equality follows from E(f ) = 0. βi is the vector with factor loadings,

βi≡ E(f f0)−1E(f Ri). (67)

Further rewriting gives

E(R) = 1

a−

E(Rf0)E(f f0)−1E(f f0)b

a =

1 a− β

0E(f f0)b

a . (68)

Lastly define γ and λ

γ ≡ 1 E(m) = 1 a, (69) λ ≡ −1 aE(f f 0_)b, ₍₇₀₎ to find E(R) = γ + β0λ. (71)

8.2 Linear Factor Model implies an Expected Return-Beta Model in

APT

First the concept of asymptotic arbitrage is defined to show that the linear factor model implies a linear pricing model.

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θn_{= {{θ}2 1, θ22}, {θ31, θ23, θ33}, ..., {θn1, θ2n, ..., θnn}} such that: n X i=1 θ_in= 0, n X i=1 θinE(Ri) ≥ η > 0, n X i=1 n X j=1 θn_iθ_jnσij P −→ 0, (72)

where n represents the number of currencies in a portfolio and η is a constant.

If returns are generated by the factor model in equation (19) and equation (20) and there are no asymptotic arbitrage opportunities, then for all t there exists a linear pricing model such that

νi = µRi− λ0− k X j=1 λjβij, lim n→∞ 1 n n X i=1 νi2= 0, (73)

where νiis the error term of the regression of µR onto a vector of ones and the matrix β. The OLS errors are orthogonal to the regression, so the following conditions hold:

n X i=1 νi = 0, (74) n X i=1 νiβik= 0, ∀k. (75)

To show that asymptotic arbitrage should not be possible, the following zero cost portfolio is considered: θi= νi pn Pn i=1ν 2 i = ψνi, (76) where ψ = √ 1 nPn i=1νi2

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n X i=1 θiri= ψ n X i=1 νiri (77) = ψ n X i=1 νi[µRi+ k X j=1 βijfj+ εi] (78) = ψ n X i=1 νi(µRi+ εi). (79)

The expected profit of this portfolio is:

E ψ n X i=1 νi(µRi+ εi) = ψ n X i=1 νiµRi (80) = ψ λ0 n X i=1 νi+ k X j=1 λj n X i=1 νiβij+ n X i=1 ν_i2 (81) = v u u t 1 n n X i=1 ν2 i. (82)

The variance of the profit is:

V[ψ n X i=1 νi(µRi+ εi)] = ψ 2 n X i=1 ν_i2σ2_i (83) = Pn i=1ν 2 iσ2i nPn i=1ν 2 i ≤σ 2 n, (84) where σ2_{= max{σ}2

1, σ22, ..., σ2n}. The variance goes to zero as n grows larger, which implies that the expected profit should also go to zero in equilibrium such that asymptotic arbitrage opportunities are not possible, so

Pn i=1ν

2 i

n → 0 as n grows larger. If large well diversified currency portfolios are considered instead of individual currencies the idiosyncratic risk νi can be assumed to be zero, so in conclusion the following holds:

E(Rt) = ιNλ0+ βλF, (85)

where ιN is a N-dimensional vector with ones, λ0is the k-dimensional zero-β return vector and λF the k-dimensional vector with factor risk premia (Ross, 1976).

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8.3 Behaviour of CLR Statistic at Critical Points of the FAR Statistic

When the GLS-LM statistic is equal to zero such that JGLS(λF,0) = FAR(λF,0), then the CLR statistic can be written as:

CLR(λF,0) = 1 2 FAR(λF,0) − r(λF,0) + v u u t FAR(λF,0) + r(λF,0) !2 − 4r(λF,0)FAR(λF,0) ! (86) =1 2 FAR(λF,0) − r(λF,0) + q FAR(λF,0)2− 2r(λF,0)FAR(λF,0) + r(λF,0)2 ! . (87) (88) If FAR(λF,0) > r(λF,0) then the CLR becomes

CLR(λF,0) = 1 2 FAR(λF,0) − r(λF,0) + q FAR(λF,0)2− 2r(λF,0)FAR(λF,0) + r(λF,0)2 ! (89) = 1 2 FAR(λF,0) − r(λF,0) + FAR(λF,0) − r(λF,0) ! . (90) = FAR(λF,0) − r(λF,0) (91)

If FAR(λF,0) < r(λF,0) then the CLR reduces to

CLR(λF,0) = 1 2 FAR(λF,0) − r(λF,0) − FAR(λF,0) − r(λF,0) ! = 0. (92)

8.4 Matlab code of pca r2 mc simulation.m

%% Load data

load data_all_lustig_FGR.mat % Rescale data if necessary

% % CCAPM % % G = G(:,1);

% % DCAPM

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% %EZ-CCAPM

% G = [G(:,1) G(:,end)];

[T, m] = size(G); [~, N] = size(R);

%% PCA on portfolio returns V_RR = cov(R);

[COEFF_pf,latent_pf,explained_pf] = pcacov(V_RR);

% Fama-MacBeth 2 Stage Regression

% First Stage: regress portfolio returns on factors and a constant coeff_1st = [ones(T,1) G]\R;

mu_hat = coeff_1st(1,:)’; B_hat = coeff_1st(2:end,:)’;

% Second Stage: regress average returns on estimated beta’s and a const. R_avg = mean(R)’;

coeff_2nd = [ones(N,1) B_hat]\R_avg; factor_prices = coeff_2nd(2:end);

% FM: covariance matrix of factor prices X = [ones(N,1) B_hat]; invXX = inv(X’*X); Xlambda = X*coeff_2nd; err = R - repmat(Xlambda’,T,1); Sigma = 1/T*(err’)*err; cov_factor_prices = 1/T*invXX*X’*Sigma*X*invXX; % Print results: for i=1:m fprintf(’Factor price %d: %.4f (%.4f) \n’,... i, factor_prices(i), sqrt(cov_factor_prices(i+1,i+1))); end

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% Compute residuals and apply PCA on covariance matrix of residuals % res = R - repmat(mu_hat’,T,1) - G*B_hat’;

res = R - [ones(T,1) G]*coeff_1st; V_res = cov(res); [COEFF_res,latent_res,explained_res] = pcacov(V_res); %% LR-test H0: B=0 LR = T*sum(latent_pf - latent_res); p_val = 1-chi2cdf(LR,3*N); % Separate LR-tests % Non-durable X = [ones(T,1) G(:,2:end)];

[LR_nd, p_val_nd] = LR_test(X, R, latent_res, T, N);

% Durable

X = [ones(T,1) G(:,1) G(:,end)];

[LR_d, p_val_d] = LR_test(X, R, latent_res, T, N);

% Market

X = [ones(T,1) G(:,1:2)];

[LR_m, p_val_m] = LR_test(X, R, latent_res, T, N);

%% pseudo R2

pseudo_R2 = 1 - sum(latent_res)/sum(latent_pf);

%% Using Lustig, Roussanov and Verdelhan (2011) factors RX and HML_FX

% FM: F on R % 1st stage

coeff_F = [ones(T,1) F]\R; B_hat_F = coeff_F(2:end,:)’;

% 2nd stage

coeff_F_2nd = [ones(N,1) B_hat_F]\R_avg; mu_hat_F = [ones(N,1) B_hat_F]*coeff_F_2nd;

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factor_prices_F = coeff_F_2nd(2:end);

% FM: covariance matrix of factor prices for F X = [ones(N,1) B_hat_F]; invXX = inv(X’*X); Xlambda = X*coeff_F_2nd; err = R - repmat(Xlambda’,T,1); Sigma = 1/T*(err’)*err; cov_factor_prices_F = 1/T*invXX*X’*Sigma*X*invXX; % Print results: for i=1:k

fprintf(’Factor price %d: %.4f (%.4f) \n’, i, factor_prices_F(i),... sqrt(cov_factor_prices_F(i+1,i+1)));

end

% PCA

res_F = R - [ones(T,1) F]*coeff_F; V_res_F = cov(res_F); [COEFF_res_F,latent_res_F,explained_res_F] = pcacov(V_res_F); % PCA Statistics pseudo_R2_F = 1 - sum(latent_res_F)/sum(latent_pf); LR_F = T*sum(latent_pf - latent_res_F); % LR separate % Level X = [ones(T,1) F(:,2)];

[LR_level, p_val_level] = LR_test(X, R, latent_res_F, T, N);

% HML_FX

X = [ones(T,1) F(:,1)];

[LR_HML, p_val_HML] = LR_test(X, R, latent_res_F, T, N);

%% Simulation R-squared

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[~, k] = size(F);

mc_reps = 10000;

M_i = eye(N) - ones(N,N)/N;

R_sq_OLS_results = zeros(mc_reps,2); FACCHECK_results = zeros(mc_reps,2);

% Select number of observed factors (G) to include num_fac = 3;

for i=1:num_fac;

X = [F(:,2) G(:,1:(i-1))]; for rep = 1:mc_reps

err = mvnrnd(zeros(1,N),V_res_F, T); F_rep = F;

R_rep = repmat(mu_hat_F’,T,1) + F_rep*coeff_F(2:end,:) + err;

coeff_1st_rep = [ones(T,1) X]\R_rep; B_hat_rep = coeff_1st_rep(2:end,:)’;

res_rep = R_rep - [ones(T,1) X]*coeff_1st_rep; V_res_rep = cov(res_rep);

[~, ~, explained_res_rep] = pcacov(V_res_rep); % First two principal components

FACCHECK_results(rep,i) = sum(explained_res_rep(1:2))/100;

coeff_2nd_rep = [ones(N,1) B_hat_rep]\R_avg_rep; Xb = [ones(N,1) B_hat_rep]*coeff_2nd_rep;

R_sq_OLS_rep = (Xb’*M_i*Xb)/(R_avg_rep’*M_i*R_avg_rep);

R_sq_OLS_results(rep,i) = R_sq_OLS_rep;

end end

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f = zeros(num_fac, 100); x = f;

g = f; y = f;

for i=1:num_fac

[f(i,:),x(i,:)] = ksdensity(R_sq_OLS_results(:,i), ’support’, [0,1]); [g(i,:),y(i,:)] = ksdensity(FACCHECK_results(:,i), ’support’, [0,1]); end % subplot(1,2,1) figure plot(x(1,:),f(2,:),’--’,... x(2,:),f(2,:),’-.’,... x(3,:),f(3,:)... );

xlabel(’R^2_{OLS}’, ’interpreter’, ’tex’) ylabel(’Density Function’) % subplot(1,2,2) figure plot(y(1,:),g(2,:),’--’,... y(2,:),g(2,:),’-.’,... y(3,:),g(3,:)... ); xlabel(’FACCHECK’) ylabel(’Density Function’)

8.5 Matlabe code of LR test.m

function [LR, p_val] = LR_test(X, R, latent_res, T, N) % Perform LR-test H0: B=0 using principal components % Regress R on other factors and compute residuals coeff = X\R;

Weak identification in linear factor models