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Monte Carlo simulation study

This study checks that the rewrite (2.0.0) estimates the ARMA-GARCH/APARCH models with stable and GEV innovations at least as accurately as the 2012 version (do Rego Sousa, master thesis), and adds the consistency and standard-error calibration evidence the thesis did not report. The reproducible scripts live in dev/benchmarks/arma-garch-mc/; the figures below are built from precomputed results (the study takes hours and cannot run during a package build).

Part A — non-degradation against the 2012 thesis

Every maximum-likelihood simulation table of the thesis appendix (Tables 9.2, 9.7, 9.11–9.17) is reproduced with 2.0.0: 100 series of length n = 2500, the same parameter sets. For each parameter we compare the new RMSE to the 2012 RMSE. The key picture: plot the two against each other. A point on or below the diagonal is as accurate as, or better than, 2012.

Most points sit on or below the diagonal. Summarised by how the ratio rmse_new / rmse_2012 falls (ties are within Monte Carlo error at R = 100):

The stable family (the package’s headline) has no regressions; the GEV family is median-unbiased, with a few higher-variance parameters in the ARMA(2,2) cells — which the next section shows are a small-sample effect.

Full per-parameter table (173 rows)
Cell Param True RMSE 2.0.0 RMSE 2012 Ratio Comparable Pass
9.11:set1 alpha1 0.10 0.0235 0.0648 0.36 TRUE TRUE
9.11:set1 alpha2 0.10 0.0236 0.0620 0.38 TRUE TRUE
9.11:set1 ar1 0.50 0.0262 0.0246 1.07 TRUE TRUE
9.11:set1 ar2 0.40 0.0248 0.0231 1.07 TRUE TRUE
9.11:set1 beta1 0.10 0.0923 0.0893 1.03 TRUE TRUE
9.11:set1 beta2 0.05 0.0453 0.0479 0.95 TRUE TRUE
9.11:set1 delta 1.30 0.1975 0.2229 0.89 TRUE TRUE
9.11:set1 gamma1 0.50 0.1611 0.1602 1.01 TRUE TRUE
9.11:set1 gamma2 -0.50 0.1893 0.1837 1.03 TRUE TRUE
9.11:set1 ma1 -0.40 0.0264 0.0235 1.12 TRUE FALSE
9.11:set1 ma2 0.40 0.0192 0.0197 0.98 TRUE TRUE
9.11:set1 mu 0.10 0.0077 0.0095 0.80 FALSE TRUE
9.11:set1 omega 0.05 0.0220 0.0438 0.50 FALSE TRUE
9.11:set1 stable_alpha 1.80 0.0266 0.0289 0.92 TRUE TRUE
9.11:set1 stable_beta -0.70 0.1074 0.0931 1.15 TRUE FALSE
9.12:set1 alpha1 0.05 0.0192 0.0313 0.61 TRUE TRUE
9.12:set1 alpha2 0.10 0.0221 0.0534 0.41 TRUE TRUE
9.12:set1 ar1 0.20 0.0387 0.0727 0.53 TRUE TRUE
9.12:set1 ar2 -0.20 0.0309 0.0469 0.66 TRUE TRUE
9.12:set1 beta1 0.05 0.0768 0.0844 0.91 TRUE TRUE
9.12:set1 beta2 0.20 0.0800 0.0970 0.82 TRUE TRUE
9.12:set1 delta 1.10 0.1770 0.2479 0.71 TRUE TRUE
9.12:set1 gamma1 0.20 0.2463 0.2498 0.99 TRUE TRUE
9.12:set1 gamma2 -0.20 0.1346 0.1325 1.02 TRUE TRUE
9.12:set1 ma1 0.30 0.0345 0.0938 0.37 TRUE TRUE
9.12:set1 ma2 0.50 0.0166 0.0284 0.58 TRUE TRUE
9.12:set1 mu 0.00 0.0235 0.1749 0.13 FALSE TRUE
9.12:set1 omega 0.20 0.0374 0.1176 0.32 FALSE TRUE
9.12:set1 stable_alpha 1.60 0.0281 0.0387 0.73 TRUE TRUE
9.12:set1 stable_beta 0.50 0.0691 0.0834 0.83 TRUE TRUE
9.13:set1 alpha1 0.10 0.0186 0.0483 0.39 TRUE TRUE
9.13:set1 alpha2 0.10 0.0234 0.0487 0.48 TRUE TRUE
9.13:set1 ar1 0.20 0.0539 0.0457 1.18 TRUE FALSE
9.13:set1 ar2 -0.20 0.0397 0.0392 1.01 TRUE TRUE
9.13:set1 beta1 0.10 0.1317 0.1254 1.05 TRUE TRUE
9.13:set1 beta2 0.10 0.0856 0.0967 0.89 TRUE TRUE
9.13:set1 ma1 0.30 0.0483 0.0446 1.08 TRUE TRUE
9.13:set1 ma2 0.50 0.0237 0.0265 0.90 TRUE TRUE
9.13:set1 mu 0.10 0.0106 0.0455 0.23 FALSE TRUE
9.13:set1 omega 0.10 0.0164 0.0452 0.36 FALSE TRUE
9.13:set1 stable_alpha 1.90 0.0230 0.0227 1.02 TRUE TRUE
9.13:set1 stable_beta 0.00 0.2002 0.2038 0.98 TRUE TRUE
9.13:set2 alpha1 0.10 0.0175 0.0428 0.41 TRUE TRUE
9.13:set2 alpha2 0.10 0.0219 0.0498 0.44 TRUE TRUE
9.13:set2 ar1 0.20 0.0429 0.0401 1.07 TRUE TRUE
9.13:set2 ar2 -0.20 0.0296 0.0376 0.79 TRUE TRUE
9.13:set2 beta1 0.10 0.1182 0.1192 0.99 TRUE TRUE
9.13:set2 beta2 0.10 0.0810 0.0783 1.04 TRUE TRUE
9.13:set2 ma1 0.30 0.0377 0.0397 0.95 TRUE TRUE
9.13:set2 ma2 0.50 0.0198 0.0218 0.91 TRUE TRUE
9.13:set2 mu 0.10 0.0109 0.0583 0.19 FALSE TRUE
9.13:set2 omega 0.10 0.0141 0.0442 0.32 FALSE TRUE
9.13:set2 stable_alpha 1.70 0.0308 0.0320 0.96 TRUE TRUE
9.13:set2 stable_beta -0.50 0.0771 0.1882 0.41 TRUE TRUE
9.14:set1 alpha1 0.10 0.0200 0.0483 0.41 TRUE TRUE
9.14:set1 ar1 0.20 0.0347 0.0457 0.76 TRUE TRUE
9.14:set1 beta1 0.10 0.1449 0.1254 1.16 TRUE FALSE
9.14:set1 ma1 0.30 0.0342 0.0446 0.77 TRUE TRUE
9.14:set1 mu 0.10 0.0064 0.0455 0.14 FALSE TRUE
9.14:set1 omega 0.10 0.0176 0.0452 0.39 FALSE TRUE
9.14:set1 stable_alpha 1.90 0.0236 0.0227 1.04 TRUE TRUE
9.14:set1 stable_beta 0.00 0.1948 0.2038 0.96 TRUE TRUE
9.14:set2 alpha1 0.20 0.0213 0.0814 0.26 TRUE TRUE
9.14:set2 ar1 0.30 0.0316 0.0341 0.93 TRUE TRUE
9.14:set2 beta1 0.15 0.0655 0.0687 0.95 TRUE TRUE
9.14:set2 ma1 0.20 0.0351 0.0329 1.07 TRUE TRUE
9.14:set2 mu 0.10 0.0067 0.0204 0.33 FALSE TRUE
9.14:set2 omega 0.10 0.0097 0.0434 0.22 FALSE TRUE
9.14:set2 stable_alpha 1.80 0.0256 0.0253 1.01 TRUE TRUE
9.14:set2 stable_beta 0.50 0.1077 0.0969 1.11 TRUE FALSE
9.15:set1 alpha1 0.10 0.0278 0.0297 0.94 TRUE TRUE
9.15:set1 alpha2 0.10 0.0337 0.0334 1.01 TRUE TRUE
9.15:set1 ar1 0.50 0.0209 0.0180 1.16 TRUE FALSE
9.15:set1 ar2 0.40 0.0204 0.0177 1.15 TRUE FALSE
9.15:set1 beta1 0.10 0.0753 0.0543 1.39 TRUE FALSE
9.15:set1 beta2 0.05 0.0466 0.0421 1.11 TRUE FALSE
9.15:set1 delta 1.30 0.1899 0.1775 1.07 TRUE TRUE
9.15:set1 gamma1 0.50 0.1588 0.2101 0.76 TRUE TRUE
9.15:set1 gamma2 -0.50 0.3358 0.3278 1.02 TRUE TRUE
9.15:set1 ma1 -0.40 0.0205 0.0162 1.27 TRUE FALSE
9.15:set1 ma2 0.40 0.0186 0.0143 1.30 TRUE FALSE
9.15:set1 mu 0.10 0.0104 0.0089 1.17 TRUE FALSE
9.15:set1 omega 0.05 0.0192 0.0165 1.16 TRUE FALSE
9.15:set1 xi 0.20 0.0177 0.0155 1.14 TRUE FALSE
9.15:set2 alpha1 0.05 0.0271 0.0258 1.05 TRUE TRUE
9.15:set2 alpha2 0.10 0.0382 0.0349 1.09 TRUE TRUE
9.15:set2 ar1 0.20 0.0444 0.0405 1.10 TRUE TRUE
9.15:set2 ar2 -0.20 0.0360 0.0347 1.04 TRUE TRUE
9.15:set2 beta1 0.05 0.0829 0.0693 1.20 TRUE FALSE
9.15:set2 beta2 0.20 0.0820 0.0765 1.07 TRUE TRUE
9.15:set2 delta 1.50 0.2840 0.3071 0.92 TRUE TRUE
9.15:set2 gamma1 0.20 0.5760 0.5504 1.05 TRUE TRUE
9.15:set2 gamma2 -0.20 0.3613 0.3325 1.09 TRUE TRUE
9.15:set2 ma1 0.30 0.0403 0.0377 1.07 TRUE TRUE
9.15:set2 ma2 0.50 0.0225 0.0189 1.19 TRUE FALSE
9.15:set2 mu 0.00 0.0240 0.0120 1.99 TRUE FALSE
9.15:set2 omega 0.20 0.0456 0.0511 0.89 TRUE TRUE
9.15:set2 xi 0.10 0.0167 0.0131 1.28 TRUE FALSE
9.16:set1 alpha1 0.05 0.0148 0.0097 1.53 TRUE FALSE
9.16:set1 alpha2 0.10 0.0215 0.0167 1.29 TRUE FALSE
9.16:set1 ar1 0.20 0.0602 0.0442 1.36 TRUE FALSE
9.16:set1 ar2 -0.20 0.0471 0.0363 1.30 TRUE FALSE
9.16:set1 beta1 0.05 0.1494 0.0799 1.87 TRUE FALSE
9.16:set1 beta2 0.20 0.1200 0.0743 1.61 TRUE FALSE
9.16:set1 ma1 0.30 0.0556 0.0380 1.46 TRUE FALSE
9.16:set1 ma2 0.50 0.0299 0.0220 1.35 TRUE FALSE
9.16:set1 mu 0.10 0.0300 0.0466 0.65 TRUE TRUE
9.16:set1 omega 0.20 0.0354 0.0223 1.59 TRUE FALSE
9.16:set1 xi -0.10 0.0138 0.2030 0.07 TRUE TRUE
9.16:set2 alpha1 0.10 0.0199 0.0191 1.04 TRUE TRUE
9.16:set2 alpha2 0.20 0.0296 0.0264 1.12 TRUE FALSE
9.16:set2 ar1 0.20 0.0635 0.0571 1.11 TRUE FALSE
9.16:set2 ar2 -0.20 0.0504 0.0474 1.06 TRUE TRUE
9.16:set2 beta1 0.10 0.0974 0.0846 1.15 TRUE FALSE
9.16:set2 beta2 0.10 0.0670 0.0617 1.09 TRUE TRUE
9.16:set2 ma1 0.30 0.0580 0.0530 1.10 TRUE TRUE
9.16:set2 ma2 0.50 0.0304 0.0245 1.24 TRUE FALSE
9.16:set2 mu 0.10 0.0222 0.0444 0.50 TRUE TRUE
9.16:set2 omega 0.10 0.0130 0.0123 1.06 TRUE TRUE
9.16:set2 xi -0.20 0.0126 0.0122 1.04 TRUE TRUE
9.17:set1 alpha1 0.20 0.0213 0.0204 1.04 TRUE TRUE
9.17:set1 ar1 0.30 0.0245 0.0221 1.11 TRUE FALSE
9.17:set1 beta1 0.20 0.0347 0.0417 0.83 TRUE TRUE
9.17:set1 ma1 0.40 0.0240 0.0201 1.19 TRUE FALSE
9.17:set1 mu 0.10 0.0121 0.0295 0.41 TRUE TRUE
9.17:set1 omega 0.05 0.0037 0.0040 0.94 TRUE TRUE
9.17:set1 xi 0.10 0.0165 0.0141 1.18 TRUE FALSE
9.17:set2 alpha1 0.20 0.0237 0.0219 1.08 TRUE TRUE
9.17:set2 ar1 0.30 0.0312 0.0266 1.17 TRUE FALSE
9.17:set2 beta1 0.20 0.0560 0.0633 0.89 TRUE TRUE
9.17:set2 ma1 0.40 0.0299 0.0242 1.23 TRUE FALSE
9.17:set2 mu 0.10 0.0124 0.0290 0.43 TRUE TRUE
9.17:set2 omega 0.05 0.0050 0.0057 0.88 TRUE TRUE
9.17:set2 xi -0.10 0.0140 0.0144 0.97 TRUE TRUE
9.2:theta1 alpha1 0.45 0.0338 0.0595 0.57 TRUE TRUE
9.2:theta1 ar1 0.32 0.0161 0.0896 0.18 TRUE TRUE
9.2:theta1 beta1 0.08 0.0206 0.0181 1.14 TRUE FALSE
9.2:theta1 mu 0.21 0.0071 0.0783 0.09 TRUE TRUE
9.2:theta1 omega 0.01 0.0006 0.0375 0.02 TRUE TRUE
9.2:theta1 xi 0.08 0.0166 0.0185 0.90 TRUE TRUE
9.2:theta2 alpha1 0.80 0.0386 0.0417 0.93 TRUE TRUE
9.2:theta2 ar1 0.10 0.0127 0.0199 0.64 TRUE TRUE
9.2:theta2 beta1 0.02 0.0055 0.0054 1.01 TRUE TRUE
9.2:theta2 mu 0.01 0.0044 0.0041 1.05 TRUE TRUE
9.2:theta2 omega 0.01 0.0021 0.0016 1.32 TRUE FALSE
9.2:theta2 xi 0.20 0.0189 0.0235 0.80 TRUE TRUE
9.2:theta3 alpha1 0.50 0.0302 0.0299 1.01 TRUE TRUE
9.2:theta3 ar1 0.20 0.0139 0.0293 0.48 TRUE TRUE
9.2:theta3 beta1 0.10 0.0118 0.0123 0.97 TRUE TRUE
9.2:theta3 mu 0.05 0.0084 0.0119 0.71 TRUE TRUE
9.2:theta3 omega 0.05 0.0037 0.0063 0.58 TRUE TRUE
9.2:theta3 xi 0.30 0.0187 0.0213 0.88 TRUE TRUE
9.7:phi1 alpha1 0.45 0.0239 0.0199 1.20 TRUE FALSE
9.7:phi1 ar1 0.32 0.0158 0.0164 0.96 TRUE TRUE
9.7:phi1 beta1 0.08 0.0275 0.0309 0.89 TRUE TRUE
9.7:phi1 mu 0.21 0.0049 0.0051 0.96 FALSE TRUE
9.7:phi1 omega 0.01 0.0006 0.0007 0.80 FALSE TRUE
9.7:phi1 stable_alpha 1.80 0.0246 0.0285 0.86 TRUE TRUE
9.7:phi1 stable_beta 0.50 0.1079 0.1173 0.92 TRUE TRUE
9.7:phi2 alpha1 0.80 0.0243 0.0701 0.35 TRUE TRUE
9.7:phi2 ar1 0.10 0.0104 0.0552 0.19 TRUE TRUE
9.7:phi2 beta1 0.02 0.0107 0.0438 0.24 TRUE TRUE
9.7:phi2 mu 0.01 0.0005 0.0008 0.66 FALSE TRUE
9.7:phi2 omega 0.01 0.0006 0.0009 0.64 FALSE TRUE
9.7:phi2 stable_alpha 1.90 0.0235 0.0342 0.69 TRUE TRUE
9.7:phi2 stable_beta 0.50 0.1705 0.2522 0.68 TRUE TRUE
9.7:phi3 alpha1 0.50 0.0250 0.0200 1.25 TRUE FALSE
9.7:phi3 ar1 0.20 0.0160 0.0153 1.05 TRUE TRUE
9.7:phi3 beta1 0.10 0.0238 0.0256 0.93 TRUE TRUE
9.7:phi3 mu 0.05 0.0028 0.0030 0.93 FALSE TRUE
9.7:phi3 omega 0.05 0.0030 0.0035 0.86 FALSE TRUE
9.7:phi3 stable_alpha 1.80 0.0254 0.0277 0.92 TRUE TRUE
9.7:phi3 stable_beta 0.50 0.1058 0.1011 1.05 TRUE TRUE

Consistency

For the GEV ARMA(2,2) cells (where Part A showed the only higher variance), RMSE is tracked across sample sizes n = 2500, 5000, 10000 (R = 100). Every parameter (one line) declines with n — the empirical face of consistency.

Part B — standard-error calibration (Wald coverage)

On a representative subset (two AR(1)-GARCH per family and one APARCH each), fits at n = 500, 1000, 2500, 5000 (R = 300) record the 95% Wald coverage. The shaded band is a reasonable [0.90, 0.98] window around the 0.95 target. Mean-equation parameters (mu, ar1, ar2) are highlighted.

Coverage is near nominal for the well-identified parameters. Two limitations stand out, both expected:

Bootstrap vs Wald for stable mean/AR parameters

For 9.7:phi1 at n = 2500, the 95% parametric-bootstrap coverage next to the Wald coverage (R = 80 replications, B = 99 resamples). The bootstrap lifts the mu/ar1 coverage back toward 0.95.

Reference

do Rego Sousa, T. (2012). Modelos combinados AR-GARCH. Master thesis, Universidade de Brasília. Appendix, Chapter 9 (MLE Tables 9.2, 9.7, 9.11–9.17).

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