Marco Filosófico

B.1 Data generating process

In this appendix, we extend the design of the single factor Monte Carlo experiment in Peñaranda and Sentana (2015) to a two-factor model. An unrestricted (i.i.d.) Gaussian data generating process (DGP) for (f ; r) is

f N ( ; ) ;

r= _r+ B_r(f ) + u_r; u_r N (0; _rr) ; where the n 2 matrix B_r is de…ned by the two beta vectors

B_r= _{1 2} :

Without loss of generality, we construct the two factors so that their covariance matrix is the identity matrix. In addition, given that we use the simulated data to test that an a¢ ne function of f is orthogonal to r, the only thing that matters is the linear span of r. As a result, we can substantially reduce the number of parameters characterizing the conditional DGP for r by means of the following steps:

1. a Cholesky transformation of r which e¤ectively sets the residual variance rr equal to the identity matrix,

2. a Householder transformation that makes the second to the last entries of the vector of risk premia _r equal to zero (see Householder (1964)),

3. another Householder transformation that makes the third to the last entries of ₁ equal to zero,

4. a …nal third Householder transformation that makes the fourth to the last entries of ₂ equal to zero.

As a result, our simpli…ed DGP for excess returns will be

r= _re1+ ( ₁₁e1+ ₂₁e2) (f1 1) + ( ₁₂e1+ ₂₂e2+ ₃₂e3) (f2 2) + ur; u_r N (0; I_n) ;

where (e₁; e₂; e₃) are the …rst, second, and third columns of the identity matrix, and

f N ( ; I2) : B.2 Model restrictions

We set the values of the three parameters of to 1. In turn, we calibrate the six parameters that de…ne r as follows. First, we de…ne a Hansen-Jagannathan (HJ) distance for this three-factor model as the minimum with respect to of the quadratic form

0M⁰V ar ¹(r) M ;

The 3 3 weighting matrix

W = M⁰V ar ¹(r) M

can be interpreted as the variance matrix of three noteworthy portfolios. The …rst one yields the maximum Sharpe ratio

r₀ = r⁰V ar ¹(r) E (r) ;

while the other two are the centred factor mimicking portfolios r_i= r⁰V ar ¹(r) Cov (r; f_i) ; i = 1; 2:

Note that if we minimize the above quadratic form subject to the symmetric normalization

0 = 1, then this HJ distance will be equal to the minimum eigenvalue of the covariance matrix W.

The …rst entry of W is the variance of r0 or, equivalently, the squared maximum Sharpe ratio. The other two diagonal entries are the variances of (r₁; r₂) or, equivalently, the R² of their respective regressions. Finally, the three di¤erent o¤-diagonal elements correspond to the covariances between these three portfolios, which we can pin down by their correlations. In this way, we have six parameters that are easy to interpret and calibrate, from which we can obtain the six parameters that our DGP requires for r.

We start from the free design and progressively add more and more constraints. In addi-tion, we can interpret the constraints that the di¤erent models impose as forcing certain linear combinations of (r0; r1; r2) with coe¢ cients (c; b1; b2) to have zero variance. We de…ne 3 designs (with some variants) indexed by the dimension of the subspace of prices of risk d.

Design d = 0: The matrix W has full rank. We need to give values to the six parameters with the interpretations mentioned before. We calibrate their values to the data as ex-plained below. The rest of designs require constraints on the matrix W, which we impose by means of small changes in that matrix.

Design d = 1: The matrix W has one rank failure de…ned by a one-dimensional subspace of vectors (c; b₁; b₂). At least one of the factors must enter the SDF to avoid risk neutrality, so we can assume that b₂6= 0. Thus, we can choose a linear combination (c ; b1; 1) with zero variance. Equivalently, we can express r2 as

r_{2 2} = c (r_{0 0}) + b₁(r_{1 1}) ; and change the last column of matrix W to

02= c 00+ b_{1 01};

12= c ₀₁+ b_{1 11};

22= c ² 00+ b₁² 11+ 2c b_{1 01}:

We keep the three parameters that de…ne the covariance matrix of (r0; r1) equal to the values they take in design d = 0. This design will have two variants: one with nonzero c

in the linear combination (c; b₁; b₂), and a second one with c = 0. In the former variant, we choose c and b₁ to keep the same ₀₂ and ₂₂ as in the design d = 0. In the second variant, we chose c = b₁ = 0, which is equivalent to an uncorrelated factor, so that

02= 12= 22= 0.

Design d = 2: The matrix W has two rank failures de…ned by a two-dimensional subspace of vectors (c; b₁; b₂). We maintain the linear combination (c ; b₁; 1) with zero variance from design d = 1, and add a second linear combination (c ; 1; 0) with zero variance.

Equivalently, we can express r1 as

r_{1 1} = c (r_{0 0}) ; and modify the matrix W accordingly

01= c 00;

11= c ² ₀₀;

with ( 02; 12; 22) satisfying the same equations as in design d = 1. We keep 00 equal to the value in design d = 0. This design will again have two variants: one with nonzero c in the linear combinations (c; b₁; b₂), and a second one with c = c = 0. In the former variant, we choose c to keep the same ₁₁as in the design d = 0. In the second variant, we have two uncorrelated factors, and hence all entries of W except 00 are equal to 0.

B.3 Numerical values

Design d = 0: We calibrate the matrix W from the data. After orthogonalizing the factors (with the market being the …rst factor), we set the correlations of (r₀; r₁; r₂) and the variances of (r₁; r₂) to values similar to those in the data. In contrast, we lower the variance of r₀, which can be interpreted as the squared maximum Sharpe ratio, so as to ensure a realistic risk-return trade o¤. Thus, the matrix that we use is

W = 0 BB B@

0:090 0:198 0:063 0:980 0

0:100 1 CC CA;

which is associated to the following DGP

r= 1:433e₁+ (6:907e₁+ 1:140e₂) (f_{1 1}) + (0:049e₁ 0:297e₂+ 0:143e₃) (f_{2 2}) + u_r:

Design d = 1: We study two variants. In the …rst one, we impose the existence of an SDF

by means of the matrix

W =

which is associated to the following DGP

r= 1:464e1+ (7:158e1+ 1:167e2) (f1 1) + ( 0:229e1 0:373e2) (f2 2) + ur: In the second one, we impose the existence of an SDF with coe¢ cients

0

by means of the matrix

W =

which is associated to the following DGP

r= 1:432e₁+ (6:914e₁+ 1:093e₂) (f_{1 1}) + u_r:

Design d = 2: We study two variants. In the …rst one, we impose the existence of two SDFs with coe¢ cients

by means of the matrix

W =

which is associated to the following DGP

r= 3:182e1+ 10:500e1(f1 1) + 1:118e1(f2 2) + ur: In the second one, we impose the existence of two SDFs with coe¢ cients

0 BB B@

c c b₁ b₁ b₂ b₂

1 CC CA=

0 BB B@

0 0

0 1

1 0

1 CC CA

by means of the matrix

W = 0 BB B@

0:090 0 0 0 0 0

1 CC CA;

which is associated to the following DGP

r= 0:300e1+ ur:

Table 1: Empirical evaluation of the model 1951-2001

One-dimensional Set Two-dimensional Set Three-dimensional Set

Market 0.200 (0.805) -3.888 0.514 (0.002) 4.793 0 0

Nondur. 24.765 (0.458) 222.902 0 (0.000) 0 115.687 0

Durables 92.229 (0.035) 0 99.333 (0.000) 0 0 121.320

Mean 0.014 ( 0.790) -0.099 0.034 (0.494) 0.852 0.421 -0.029

Criterion 20.743 (0.537) 56.687 (0.134) 215.144 (0.000)

Criterion c = 0 20.814 (0.592) 58.098 (0.151)

Note: This table displays estimates of the SDF parameters, as well as the J tests (with free and constrained SDF means) with p-values in parenthesis (). The …rst, second and third blocks of columns report the results for sets of SDFs of dimension 1, 2 and 3, respectively. We display the estimates for a particular normalization, but our results are numerically invariant to the chosen normalization because we use CU-GMM. The J tests are complemented with signi…cance tests of some SDF parameters. In particular, the p-value of the distance metric test of the null hypothesis of zero parameters is reported in parenthesis to the right of the estimates. Distance metric tests are not reported when the p-value of the J test is lower than0:01. The payo¤s of the test assets correspond to 25 nominal excess returns of size and book-to-market sorted portfolios on a quarterly basis.

Table 2: Empirical evaluation of the model 1949-2012

One-dimensional Set Two-dimensional Set Three-dimensional Set Panel A. 25 size and book-to-market sorted portfolios

Market 0.766 (0.673) -1.878 0.882 (0.000) 12.882 0 0

Nondur. -6.452 (0.834) 192.583 0 (0.000) 0 169.191 0

Durables 106.144 (0.024) 0 97.810 (0.000) 0 0 110.143

Mean 0.003 (0.972) 0.052 0.008 (0.757) 0.411 0.065 -0.075

Criterion 18.278 (0.689) 54.818 (0.175) 165.053 (0.000)

Criterion c = 0 18.279 (0.742) 55.375 (0.216) Panel B. 6 size and book-to-market sorted portfolios

Market 0.504 (0.542) -1.873 0.597 (0.053) 4.148 0 0

Nondur. 12.831 (0.837) 208.870 0 (0.000) 0 152.818 0

Durables 92.619 (0.158) 0 100.563 (0.000) 0 0 121.767

Mean 0.021 (0.836) 0.076 0.007 (0.810) 0.875 0.326 -0.173

Criterion 0.526 (0.913) 3.787 (0.876) 47.695 (0.000)

Criterion c = 0 0.568 (0.967) 4.207 (0.938)

Note: This table displays estimates of the SDF parameters, as well as the J tests (with free and constrained SDF means) with p-values in parenthesis (). The …rst, second and third blocks of columns report the results for sets of SDFs of dimension 1, 2 and 3, respectively. We display the estimates for a particular normalization, but our results are numerically invariant to the chosen normalization because we use CU GMM. The J tests are complemented with signi…cance tests of some SDF parameters. In particular, the p-value of the distance metric test of the null hypothesis of zero parameters is reported in parenthesis to the right of the estimates. Distance metric tests are not reported when the p-value of the J test is lower than0:01. The payo¤s of the test assets correspond to 25 (Panel A) and 6 (Panel B) real excess returns of size and book-to-market sorted portfolios at the quarterly frequency.

Table 3: Rejection rates for a two-dimensional set of admissible SDFs (T = 200) Nominal size

10 5 1

Panel A. Some SDFs have nonzero mean

J d=2 13.65 7.03 1.61

J d=2, c=0 99.62 99.62 99.62

DM c=0 99.62 99.62 99.62

J d=1 1.03 0.37 0.01

J d=1 c=0 11.78 3.07 1.20

DM c=0 39.97 28.01 10.89

Panel B. All SDFs have zero mean

J d=2 8.97 4.49 0.71

J d=2, c=0 14.26 7.73 1.72

DM c=0 21.46 13.69 4.34

J d=1 0.75 0.15 0.00

J d=1, c=0 0.89 0.31 0.00

DM c=0 8.03 3.37 0.30

Note: This table displays the rejection rates of CU J tests, their variants restricted to zero SDF means, and the corresponding DM tests, as described in Section 3. The rates are shown in percentage for the asymptotic critical values at 10, 5, and 1%. 10000 samples of 6 excess returns are simulated under the two variants of a two-dimensional set of admissible SDFs in Appendix B, where the parameter values of the DGP are explained. Panel A reports the results for the …rst variant, when most of these SDFs have nonzero means, and Panel B reports the results for the second variant, when the asset pricing model is completely overspeci…ed.

Table 4: Rejection rates for a one-dimensional set of admissible SDFs (T = 200) Nominal size

10 5 1

Panel A. Some SDFs have nonzero mean

J d=2 99.17 98.03 92.79

J d=2, c=0 99.99 99.99 99.99

DM c=0 99.97 99.97 99.97

J d=1 10.06 4.99 0.98

J d=1 c=0 97.92 95.39 84.25

DM c=0 99.24 98.71 95.57

Panel B. All SDFs have zero mean

J d=2 68.55 56.95 33.30

J d=2, c=0 99.85 99.85 99.85

DM c=0 99.84 99.84 99.84

J d=1 6.29 2.66 0.33

J d=1, c=0 11.65 5.96 1.20

DM c=0 20.51 12.92 3.88

Note: This table displays the rejection rates of CU J tests, their variants restricted to zero SDF means, and the corresponding DM tests, as described in Section 3. The rates are shown in percentage for the asymptotic critical values at 10, 5, and 1%. 10000 samples of 6 excess returns are simulated under the two variants of a one-dimensional set of admissible SDFs in Appendix B, where the parameter values of the DGP are explained. Panel A reports the results for the …rst variant, when the SDF has a nonzero mean, and Panel B reports the results for the second variant, when the asset pricing model is overspeci…ed.

Table B1: Rejection rates for a two-dimensional set of admissible SDFs (T = 600) Nominal size

10 5 1

Panel A. Some SDFs have nonzero mean

J d=2 10.80 5.55 1.13

J d=2, c=0 99.55 99.55 99.55

DM c=0 99.55 99.55 99.55

J d=1 0.78 0.16 0.00

J d=1 c=0 10.36 5.20 1.23

DM c=0 37.94 26.15 9.82

Panel B. All SDFs have zero mean

J d=2 10.15 5.03 0.99

J d=2, c=0 11.58 5.98 1.28

DM c=0 13.63 7.58 1.63

J d=1 0.77 0.16 0.00

J d=1, c=0 0.82 0.13 0.00

DM c=0 5.94 2.09 0.19

Note: This table displays the rejection rates of CU J tests, their variants restricted to zero SDF means, and the corresponding DM tests, as described in Section 3. The rates are shown in percentage for the asymptotic critical values at 10, 5, and 1%. 10000 samples of 6 excess returns are simulated under the two variants of a two-dimensional set of admissible SDFs in Appendix B, where the parameter values of the DGP are explained. Panel A reports the results for the …rst variant, when most of these SDFs have nonzero means, and Panel B reports the results for the second variant, when the asset pricing model is completely overspeci…ed.

Table B2: Rejection rates for a one-dimensional set of admissible SDFs (T = 600) Nominal size

10 5 1

Panel A. Some SDFs have nonzero mean

J d=2 100 100 100

J d=2, c=0 100 100 100

DM c=0 100 100 100

J d=1 9.94 5.00 1.06

J d=1 c=0 100 100 100

DM c=0 100 100 100

Panel B. All SDFs have zero mean

J d=2 99.06 97.97 92.69

J d=2, c=0 100.00 100.00 100.00 DM c=0 100.00 100.00 100.00

J d=1 8.87 4.26 0.82

J d=1, c=0 10.50 5.25 1.11

DM c=0 12.79 6.63 1.60

Note: This table displays the rejection rates of CU J tests, their variants restricted to zero SDF means, and the corresponding DM tests, as described in Section 3. The rates are shown in percentage for the asymptotic critical values at 10, 5, and 1%. 10000 samples of 6 excess returns are simulated under the two variants of a one-dimensional set of admissible SDFs in Appendix B, where the parameter values of the DGP are explained. Panel A reports the results for the …rst variant, when the SDF has a nonzero mean, and Panel B reports the results for the second variant, when the asset pricing model is overspeci…ed.

Figure 1: One asset

Figure 2: Two assets

Figure 3: Three segmented asset markets

Figure 4: Three integrated asset markets

Figure 5: Normalizations

Figure 6: Valid but unattractive model with a useless factor

Figure 7: Valid and attractive model with a useless factor

Figure 8: Two single factor models

Figure 9: Two useless factors

Figure 10: An unpriced second factor

Figure 11: Risk premia from 2S-GMM

1 1.5 2 2.5 3 3.5 4

1 1.5 2 2.5 3 3.5 4

2S-GMM

model (%)

data (%)

Figure 12: Risk premia from IT-GMM

Figure 13: Risk premia from IT-GMM, free coe¢ cients

1 1.5 2 2.5 3 3.5 4

In document PLAN DE DESARROLLO COMUNITARIO VEREDA EL GARZON VEREDA EL GARZÓN EL GARZÓN: UNA EMPRESA PRODUCTIVA, SOSTENIBLE Y SOCIALMENTE JUSTA. (página 119-127)