------------------------------------------------------------------------------- Variable Mean Std Dev Variance Minimum Maximum Valid Missing ------------------------------------------------------------------------------- Z_1 0.0101 0.0859 0.0074 -0.2708 0.1540 60 0 Z_2 0.0043 0.1124 0.0126 -0.3009 0.3237 60 0 Z_3 0.0102 0.0642 0.0041 -0.1669 0.1470 60 0 Z_4 0.0083 0.0998 0.0100 -0.2443 0.2622 60 0 Z_5 0.0188 0.0572 0.0033 -0.1265 0.2130 60 0 Z_6 -0.0017 0.1067 0.0114 -0.2707 0.2242 60 0 Z_7 0.0227 0.0678 0.0046 -0.1476 0.1935 60 0 Z_8 0.0061 0.0497 0.0025 -0.1219 0.1089 60 0 Z_9 0.0101 0.0711 0.0051 -0.1611 0.1949 60 0 Z_10 0.0058 0.0509 0.0026 -0.1267 0.1255 60 0 Z_11 0.0252 0.1184 0.0140 -0.2366 0.4112 60 0 Z_12 0.0062 0.0581 0.0034 -0.1273 0.1528 60 0 Z_13 0.0122 0.0912 0.0083 -0.2390 0.2700 60 0 Z_14 0.0068 0.0903 0.0082 -0.1620 0.2449 60 0 Z_15 0.0143 0.0509 0.0026 -0.1030 0.1533 60 0 Z_16 0.0132 0.0655 0.0043 -0.1539 0.1331 60 0 Z_17 0.0118 0.0622 0.0039 -0.1395 0.1241 60 0 Z_18 0.0082 0.0691 0.0048 -0.1704 0.1621 60 0 Z_19 0.0059 0.0373 0.0014 -0.1079 0.0877 60 0 Z_20 0.0079 0.0433 0.0019 -0.0995 0.1274 60 0 Z_DJ 0.0060 0.0411 0.0017 -0.1111 0.0941 60 0 =========================================================================== ================== Estimates with Intercepts ============================= Valid cases: 60 Dependent variable: Z_1 Missing cases: 0 Deletion method: None Total SS: 0.436 Degrees of freedom: 58 R-squared: 0.630 Rbar-squared: 0.624 Residual SS: 0.161 Std error of est: 0.053 F(1,58): 98.853 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT 0.000108 0.006876 0.015680 0.988 --- --- Z_DJ 1.659884 0.166948 9.942506 0.000 0.793869 0.793869 =============================================================================== Valid cases: 60 Dependent variable: Z_2 Missing cases: 0 Deletion method: None Total SS: 0.746 Degrees of freedom: 58 R-squared: 0.312 Rbar-squared: 0.301 Residual SS: 0.513 Std error of est: 0.094 F(1,58): 26.349 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT -0.004908 0.012269 -0.400039 0.691 --- --- Z_DJ 1.529015 0.297874 5.133088 0.000 0.558907 0.558907 =============================================================================== Valid cases: 60 Dependent variable: Z_3 Missing cases: 0 Deletion method: None Total SS: 0.243 Degrees of freedom: 58 R-squared: 0.648 Rbar-squared: 0.642 Residual SS: 0.085 Std error of est: 0.038 F(1,58): 106.944 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT 0.002638 0.005009 0.526573 0.600 --- --- Z_DJ 1.257746 0.121623 10.341389 0.000 0.805212 0.805212 =============================================================================== Valid cases: 60 Dependent variable: Z_4 Missing cases: 0 Deletion method: None Total SS: 0.588 Degrees of freedom: 58 R-squared: 0.579 Rbar-squared: 0.572 Residual SS: 0.248 Std error of est: 0.065 F(1,58): 79.795 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT -0.002829 0.008526 -0.331798 0.741 --- --- Z_DJ 1.849121 0.207003 8.932826 0.000 0.760977 0.760977 =============================================================================== Valid cases: 60 Dependent variable: Z_5 Missing cases: 0 Deletion method: None Total SS: 0.193 Degrees of freedom: 58 R-squared: 0.568 Rbar-squared: 0.561 Residual SS: 0.083 Std error of est: 0.038 F(1,58): 76.402 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT 0.012472 0.004942 2.523480 0.014 --- --- Z_DJ 1.048802 0.119989 8.740828 0.000 0.753962 0.753962 =============================================================================== Valid cases: 60 Dependent variable: Z_6 Missing cases: 0 Deletion method: None Total SS: 0.672 Degrees of freedom: 58 R-squared: 0.413 Rbar-squared: 0.403 Residual SS: 0.394 Std error of est: 0.082 F(1,58): 40.782 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT -0.011770 0.010762 -1.093667 0.279 --- --- Z_DJ 1.668637 0.261293 6.386086 0.000 0.642533 0.642533 =============================================================================== Valid cases: 60 Dependent variable: Z_7 Missing cases: 0 Deletion method: None Total SS: 0.271 Degrees of freedom: 58 R-squared: 0.118 Rbar-squared: 0.103 Residual SS: 0.239 Std error of est: 0.064 F(1,58): 7.796 Probability of F: 0.007 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT 0.019331 0.008374 2.308452 0.025 --- --- Z_DJ 0.567671 0.203313 2.792110 0.007 0.344218 0.344218 =============================================================================== Valid cases: 60 Dependent variable: Z_8 Missing cases: 0 Deletion method: None Total SS: 0.146 Degrees of freedom: 58 R-squared: 0.372 Rbar-squared: 0.362 Residual SS: 0.092 Std error of est: 0.040 F(1,58): 34.411 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT 0.001645 0.005185 0.317187 0.752 --- --- Z_DJ 0.738472 0.125888 5.866088 0.000 0.610220 0.610220 =============================================================================== Valid cases: 60 Dependent variable: Z_9 Missing cases: 0 Deletion method: None Total SS: 0.298 Degrees of freedom: 58 R-squared: 0.368 Rbar-squared: 0.357 Residual SS: 0.188 Std error of est: 0.057 F(1,58): 33.723 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT 0.003746 0.007439 0.503561 0.616 --- --- Z_DJ 1.048774 0.180601 5.807120 0.000 0.606349 0.606349 =============================================================================== Valid cases: 60 Dependent variable: Z_10 Missing cases: 0 Deletion method: None Total SS: 0.153 Degrees of freedom: 58 R-squared: 0.616 Rbar-squared: 0.610 Residual SS: 0.059 Std error of est: 0.032 F(1,58): 93.224 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT -0.000087 0.004149 -0.020971 0.983 --- --- Z_DJ 0.972596 0.100732 9.655247 0.000 0.785151 0.785151 =============================================================================== Valid cases: 60 Dependent variable: Z_11 Missing cases: 0 Deletion method: None Total SS: 0.827 Degrees of freedom: 58 R-squared: 0.347 Rbar-squared: 0.336 Residual SS: 0.540 Std error of est: 0.096 F(1,58): 30.828 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT 0.014953 0.012591 1.187641 0.240 --- --- Z_DJ 1.697256 0.305685 5.552307 0.000 0.589112 0.589112 =============================================================================== Valid cases: 60 Dependent variable: Z_12 Missing cases: 0 Deletion method: None Total SS: 0.199 Degrees of freedom: 58 R-squared: 0.618 Rbar-squared: 0.612 Residual SS: 0.076 Std error of est: 0.036 F(1,58): 93.908 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT -0.000462 0.004723 -0.097795 0.922 --- --- Z_DJ 1.111288 0.114677 9.690593 0.000 0.786250 0.786250 =============================================================================== Valid cases: 60 Dependent variable: Z_13 Missing cases: 0 Deletion method: None Total SS: 0.491 Degrees of freedom: 58 R-squared: 0.457 Rbar-squared: 0.448 Residual SS: 0.266 Std error of est: 0.068 F(1,58): 48.872 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT 0.003142 0.008845 0.355188 0.724 --- --- Z_DJ 1.501283 0.214750 6.990848 0.000 0.676236 0.676236 =============================================================================== Valid cases: 60 Dependent variable: Z_14 Missing cases: 0 Deletion method: None Total SS: 0.481 Degrees of freedom: 58 R-squared: 0.387 Rbar-squared: 0.377 Residual SS: 0.295 Std error of est: 0.071 F(1,58): 36.640 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT -0.001446 0.009302 -0.155401 0.877 --- --- Z_DJ 1.367033 0.225840 6.053104 0.000 0.622215 0.622215 =============================================================================== Valid cases: 60 Dependent variable: Z_15 Missing cases: 0 Deletion method: None Total SS: 0.153 Degrees of freedom: 58 R-squared: 0.375 Rbar-squared: 0.365 Residual SS: 0.096 Std error of est: 0.041 F(1,58): 34.860 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT 0.009726 0.005299 1.835331 0.072 --- --- Z_DJ 0.759636 0.128660 5.904216 0.000 0.612701 0.612701 =============================================================================== Valid cases: 60 Dependent variable: Z_16 Missing cases: 0 Deletion method: None Total SS: 0.253 Degrees of freedom: 58 R-squared: 0.522 Rbar-squared: 0.514 Residual SS: 0.121 Std error of est: 0.046 F(1,58): 63.431 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT 0.006310 0.005958 1.059158 0.294 --- --- Z_DJ 1.152014 0.144646 7.964371 0.000 0.722747 0.722747 =============================================================================== Valid cases: 60 Dependent variable: Z_17 Missing cases: 0 Deletion method: None Total SS: 0.229 Degrees of freedom: 58 R-squared: 0.493 Rbar-squared: 0.485 Residual SS: 0.116 Std error of est: 0.045 F(1,58): 56.501 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT 0.005412 0.005830 0.928305 0.357 --- --- Z_DJ 1.063897 0.141538 7.516697 0.000 0.702462 0.702462 =============================================================================== Valid cases: 60 Dependent variable: Z_18 Missing cases: 0 Deletion method: None Total SS: 0.281 Degrees of freedom: 58 R-squared: 0.707 Rbar-squared: 0.702 Residual SS: 0.082 Std error of est: 0.038 F(1,58): 140.198 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT -0.000343 0.004917 -0.069787 0.945 --- --- Z_DJ 1.413485 0.119377 11.840520 0.000 0.841049 0.841049 =============================================================================== Valid cases: 60 Dependent variable: Z_19 Missing cases: 0 Deletion method: None Total SS: 0.082 Degrees of freedom: 58 R-squared: 0.312 Rbar-squared: 0.300 Residual SS: 0.056 Std error of est: 0.031 F(1,58): 26.284 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT 0.002865 0.004072 0.703537 0.485 --- --- Z_DJ 0.506905 0.098873 5.126830 0.000 0.558438 0.558438 =============================================================================== Valid cases: 60 Dependent variable: Z_20 Missing cases: 0 Deletion method: None Total SS: 0.111 Degrees of freedom: 58 R-squared: 0.344 Rbar-squared: 0.333 Residual SS: 0.073 Std error of est: 0.035 F(1,58): 30.467 Probability of F: 0.000 Standard Prob Standardized Cor with Variable Estimate Error t-value >|t| Estimate Dep Var ------------------------------------------------------------------------------- CONSTANT 0.004199 0.004619 0.909017 0.367 --- --- Z_DJ 0.618965 0.112137 5.519701 0.000 0.586847 0.586847 =============================================================================== =============================================================================== =================== SURE ESTIMATION of the CAPM =============================== Number of Assets 20.000000 Reduced Form Parameter Estimates alpha t-alpha beta t-beta 0.00010782496 0.015680483 1.6598836 9.9425056 -0.0049081028 -0.40003910 1.5290154 5.1330883 0.0026378537 0.52657291 1.2577461 10.341389 -0.0028289744 -0.33179845 1.8491215 8.9328256 0.012471502 2.5234799 1.0488017 8.7408281 -0.011770371 -1.0936667 1.6686368 6.3860855 0.019331409 2.3084525 0.56767131 2.7921096 0.0016446696 0.31718669 0.73847228 5.8660878 0.0037458579 0.50356114 1.0487738 5.8071205 -8.7008920e-05 -0.020970877 0.97259598 9.6552466 0.014953309 1.1876411 1.6972562 5.5523065 -0.00046192258 -0.097794615 1.1112877 9.6905926 0.0031417320 0.35518807 1.5012830 6.9908484 -0.0014455458 -0.15540064 1.3670335 6.0531043 0.0097260269 1.8353308 0.75963553 5.9042162 0.0063102249 1.0591576 1.1520137 7.9643708 0.0054117942 0.92830503 1.0638968 7.5166965 -0.00034314239 -0.069787215 1.4134850 11.840520 0.0028651169 0.70353656 0.50690487 5.1268296 0.0041985591 0.90901689 0.61896458 5.5197012 Number of alphas significantly different from zero: 2.0000000 Wald Test of alpha = 0: 38.281220 p-val.: 0.0081803800 Estimated Reliability Ratio: 0.45725249 Identified Betas 3.6301247 3.3439192 2.7506600 4.0439834 2.2937037 3.6492678 1.2414833 1.6150208 2.2936426 2.1270436 3.7118578 2.4303589 3.2832691 2.9896687 1.6613043 2.5194258 2.3267162 3.0912571 1.1085885 1.3536604 Z_1 Z_2 Z_3 Z_4 Z_5 Z_6 Z_7 Z_8 Z_9 Z_10 Z_11 Z_12 Z_13 Z_14 Z_15 Z_16 Z_17 Z_18 Z_19 Z_20 =============================================================================== Minimum Distance Estimation of CAPM with Measurement Error =============================================================================== OPTMUM Version 3.1.13 11/24/2016 5:16 pm =============================================================================== return code = 0 normal convergence Value of objective function 26.121803 Parameters Estimates Gradient ----------------------------------------- Z_1 2.7954 0.0000 Z_2 2.5242 0.0000 Z_3 2.1441 0.0000 Z_4 3.0842 0.0000 Z_5 1.8922 0.0000 Z_6 2.6896 0.0000 Z_7 1.1519 0.0001 Z_8 1.2599 0.0000 Z_9 1.8036 0.0000 Z_10 1.6364 0.0000 Z_11 3.0091 0.0000 Z_12 1.8661 0.0000 Z_13 2.5592 0.0000 Z_14 2.2867 0.0000 Z_15 1.3775 0.0000 Z_16 2.0034 0.0000 Z_17 1.8460 -0.0001 Z_18 2.3760 0.0000 Z_19 0.8824 0.0000 Z_20 1.0846 0.0001 LAMBDA 0.5787 -0.0001 Number of iterations 121 Minutes to convergence 0.00363 Minimum Distance Betas 2.7954203 2.5241849 2.1441275 3.0841764 1.8922169 2.6895647 1.1519067 1.2598769 1.8035857 1.6364294 3.0090524 1.8661036 2.5592224 2.2866518 1.3775464 2.0034183 1.8459535 2.3760420 0.88243628 1.0846208 Z_1 Z_2 Z_3 Z_4 Z_5 Z_6 Z_7 Z_8 Z_9 Z_10 Z_11 Z_12 Z_13 Z_14 Z_15 Z_16 Z_17 Z_18 Z_19 Z_20 ====== Minumum Distance Parameter Estimates ================ theta std-theta t-theta 2.7954203 0.42496994 6.5779247 2.5241849 0.57643939 4.3789251 2.1441275 0.31898089 6.7218056 3.0841764 0.49482426 6.2328723 1.8922169 0.29548175 6.4038369 2.6895647 0.53501957 5.0270399 1.1519067 0.36507102 3.1552949 1.2598769 0.25542123 4.9325456 1.8035857 0.36618214 4.9253787 1.6364294 0.25210980 6.4909392 3.0090524 0.61699027 4.8769852 1.8661036 0.28727721 6.4958287 2.5592224 0.46385762 5.5172586 2.2866518 0.45993702 4.9716629 1.3775464 0.26699638 5.1594197 2.0034183 0.33358925 6.0056440 1.8459535 0.31755351 5.8130472 2.3760420 0.33760547 7.0379252 0.88243628 0.19394710 4.5498813 1.0846208 0.22511485 4.8180777 0.57874324 0.070580630 8.1997460 Minimum Distance Reliability Ratio: 0.57874324 --------------------------------------------------------- Chi-Square Test no measurement error: 12.159417 df: 1.0000000 p-val.: 0.00048840642 t-Test no measurement error: -5.9684471 p-val. 2.3952223e-09 --------------------------------------------------------- Chi-Square Test CAPM vs lin. pred. model: 26.121803 df: 19.000000 p-val.: 0.12683069 =============================================================================== Minimum Distance Estimation of CAPM with Measurement Error under restriction =============================================================================== OPTMUM Version 3.1.13 11/24/2016 5:16 pm =============================================================================== return code = 0 normal convergence Value of objective function 26.121803 Parameters Estimates Gradient ----------------------------------------- Z_1 2.7954 0.0000 Z_2 2.5242 0.0000 Z_3 2.1441 0.0000 Z_4 3.0842 0.0000 Z_5 1.8922 0.0000 Z_6 2.6896 0.0000 Z_7 1.1519 0.0000 Z_8 1.2599 0.0000 Z_9 1.8036 0.0000 Z_10 1.6364 0.0000 Z_11 3.0091 0.0000 Z_12 1.8661 0.0000 Z_13 2.5592 0.0000 Z_14 2.2867 0.0000 Z_15 1.3775 0.0000 Z_16 2.0034 0.0000 Z_17 1.8460 0.0000 Z_18 2.3760 0.0000 Z_19 0.8824 0.0000 Z_20 1.0846 0.0000 ROOT_L 0.7608 -0.0002 Number of iterations 93 Minutes to convergence 0.00285 MD (restricted) reliability ratio 0.57874341