Repository Public Research preview
1version: 12generated: '2026-11-09'3sources:4- paper/references.bib5- configs/model_registry.yaml6- configs/study.yaml7- docs/literature_matrix.md8- paper/appendices/formulas.tex9families:10 classical: Classical ARCH/GARCH and filters11 asymmetric: Asymmetric (news-impact) GARCH12 power: Power / absolute-value / Box--Cox13 fractional: Fractional persistence14 component: Component and multiplicative15 regime: Regime switching and smooth transition16 score-driven: Score-driven (GAS / DCS)17 realized: Realized-measure models and HAR18 multivariate: Multivariate19 stochastic-volatility: Stochastic volatility (parameter-driven)20 other: Other relatives (jumps, periodicity)21entries:22- id: arch23 name: ARCH($q$)24 short: ARCH25 year: 198226 family: classical27 bibkey: engle1982autoregressive28 parents: []29 equation: $\sigma_t^2=\omega+\sum_{i=1}^{q}\alpha_i\varepsilon_{t-i}^2$30 key_property: conditional variance as an autoregression in squared shocks31 stationarity: $\omega>0$, $\alpha_i\ge0$; covariance stationary iff $\sum_i\alpha_i<1$32 status_in_wp46: scored33 registry_id: arch534- id: garch35 name: GARCH($p,q$)36 short: GARCH37 year: 198638 family: classical39 bibkey: bollerslev1986generalized40 parents:41 - arch42 equation: $\sigma_t^2=\omega+\sum_{i=1}^{p}\alpha_i\varepsilon_{t-i}^2+\sum_{j=1}^{q}\beta_j\sigma_{t-j}^2$43 key_property: 'lagged conditional variances: parsimonious ARMA-type dynamics (ARCH($\infty$))'44 stationarity: $\omega>0$, $\alpha_i,\beta_j\ge0$ (sufficient); covariance stationary iff $\sum\alpha_i+\sum\beta_j<1$; strict stationarity via $\E\log(\alpha z^2+\beta)<0$45 status_in_wp46: scored46 registry_id: garch1147 note: Benchmark of WP46 (orders (1,1), (2,1), (1,2) scored). Bollerslev writes the orders as (GARCH lags, ARCH lags); arch writes p = ARCH lags, q = GARCH lags.48- id: igarch49 name: IGARCH50 short: IGARCH51 year: 198652 family: classical53 bibkey: engle1986modelling54 parents:55 - garch56 equation: $\sigma_t^2=\omega+\sum_{i}\alpha_i\varepsilon_{t-i}^2+\sum_{j}\beta_j\sigma_{t-j}^2$ with $\sum_i\alpha_i+\sum_j\beta_j=1$57 key_property: 'unit persistence: shocks to the variance never die out'58 stationarity: strictly stationary, not covariance stationary (no unconditional variance); $k$-step forecasts grow linearly in $k$59 status_in_wp46: scored60 registry_id: igarch1161- id: garch_m62 name: ARCH-M / GARCH-M63 short: GARCH-M64 year: 198765 family: classical66 bibkey: engle1987estimating67 parents:68 - garch69 equation: $r_t=\mu+\delta\,g(\sigma_t^2)+\varepsilon_t$ with $g(\sigma_t^2)\in\{\sigma_t^2,\sigma_t,\log\sigma_t^2\}$; $\sigma_t^2$ as GARCH70 key_property: conditional variance enters the conditional mean (time-varying risk premium)71 stationarity: as the underlying variance recursion72 status_in_wp46: scored73 registry_id: garchm1174 note: WP46 uses g = sigma_t^2; variance forecasts coincide with GARCH(1,1).75- id: ewma76 name: EWMA / RiskMetrics (1994)77 short: EWMA / RiskMetrics78 year: 199679 family: classical80 bibkey: jpmorgan1996riskmetrics81 parents:82 - igarch83 equation: $\sigma_t^2=(1-\lambda)\varepsilon_{t-1}^2+\lambda\sigma_{t-1}^2$, $\lambda=0.94$ (daily)84 key_property: zero-intercept IGARCH(1,1) with a fixed decay; no estimation85 stationarity: integrated; flat variance term structure86 status_in_wp46: scored87 registry_id: ewma09488 note: The verified record is the 4th edition of the Technical Document (1996); the method dates from the 1994 first release.89- id: rm200690 name: RiskMetrics 200691 short: RM200692 year: 200793 family: classical94 bibkey: zumbach2009riskmetrics95 parents:96 - ewma97 - ding_granger_lm98 equation: $\sigma_t^2=\sum_{k=1}^{K}w_k\sigma_{k,t}^2$, $\sigma_{k,t}^2=\mu_k\sigma_{k,t-1}^2+(1-\mu_k)\varepsilon_{t-1}^2$, $\mu_k=e^{-1/\tau_k}$, $\tau_k=\tau_1\rho^{k-1}$, $w_k\propto1-\log\tau_k/\log\tau_0$99 key_property: mixture of EWMAs over geometrically spaced horizons with logarithmic weights (long-memory-like kernel, no estimation)100 stationarity: integrated filter (fixed weights sum to one)101 status_in_wp46: scored102 registry_id: rm2006103 note: RiskMetrics Group technical document (2007); the verified record is the SSRN posting of 2009 (bibkey year).104- id: log_garch105 name: log-GARCH106 short: log-GARCH107 year: 1986108 family: classical109 bibkey: geweke1986comment110 reference: Geweke, J. (1986), Comment [on Engle and Bollerslev], Econometric Reviews 5(1), 57–61; Pantula, S. G. (1986), Comment, Econometric Reviews 5(1), 71–74.111 parents:112 - garch113 equation: $\log\sigma_t^2=\omega+\sum_{i=1}^{p}\alpha_i\log\varepsilon_{t-i}^2+\sum_{j=1}^{q}\beta_j\log\sigma_{t-j}^2$114 key_property: multiplicative (log-linear) variance dynamics; no positivity constraints; $\log\varepsilon_t^2$ is an ARMA115 stationarity: stationary iff the ARMA roots of $\log\varepsilon_t^2$ lie outside the unit circle; $\log\varepsilon_t^2$ undefined at zero returns116 status_in_wp46: out-of-scope117 reason: zero returns make the recursion undefined (needs ad hoc treatment); no implementation in arch 8.0.0 / rugarch 1.5-6118- id: egarch119 name: EGARCH120 short: EGARCH121 year: 1991122 family: asymmetric123 bibkey: nelson1991conditional124 parents:125 - garch126 - log_garch127 equation: $\log\sigma_t^2=\omega+\sum_{j}\beta_j\log\sigma_{t-j}^2+\sum_{i}\alpha_i\,g(z_{t-i})$, $g(z)=\theta z+\gamma(|z|-\E|z|)$128 key_property: log variance driven by sign and size of the standardized shock; no positivity constraints129 stationarity: stationary iff the roots of $1-\sum_j\beta_jL^j$ lie outside the unit circle ($|\beta|<1$ for (1,1)); $\E|z|$ depends on the innovation law130 status_in_wp46: scored131 registry_id: egarch11132 note: Parameterizations differ across Nelson, arch and rugarch (see docs/genealogy_notes.md).133- id: gjr134 name: GJR-GARCH135 short: GJR136 year: 1993137 family: asymmetric138 bibkey: glosten1993relation139 parents:140 - garch141 equation: $\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\gamma\varepsilon_{t-1}^2\ind{\varepsilon_{t-1}<0}+\beta\sigma_{t-1}^2$142 key_property: indicator on negative shocks in the variance (leverage effect, power 2)143 stationarity: $\alpha+\gamma\kappa+\beta<1$ with $\kappa=\E[z^2\ind{z<0}]$ ($=1/2$ for symmetric $z$); positivity $\alpha+\gamma\ge0$144 status_in_wp46: scored145 registry_id: gjr11146 note: Called TARCH in EViews and in several textbooks; WP46 reserves TGARCH for Zakoian's model.147- id: tgarch_zakoian148 name: TGARCH (Zakoïan)149 short: TGARCH (Zakoïan)150 year: 1994151 family: asymmetric152 bibkey: zakoian1994threshold153 parents:154 - avgarch155 equation: $\sigma_t=\omega+\alpha^{+}\varepsilon_{t-1}^{+}-\alpha^{-}\varepsilon_{t-1}^{-}+\beta\sigma_{t-1}$, $\varepsilon^{+}=\max(\varepsilon,0)$, $\varepsilon^{-}=\min(\varepsilon,0)$156 key_property: threshold on the conditional standard deviation (power 1) with separate slopes for positive and negative shocks157 stationarity: strict stationarity via $\E\log(\alpha^{+}z^{+}-\alpha^{-}z^{-}+\beta)<0$ (Zakoïan 1994); not $\alpha+\beta<1$158 status_in_wp46: scored159 registry_id: tarch11160 note: First proposed in Rabemananjara and Zakoïan (1993, JAE 8(1)) as threshold ARCH; APARCH with delta = 1 reproduces it.161- id: agarch162 name: AGARCH (Engle 1990)163 short: AGARCH164 year: 1990165 family: asymmetric166 bibkey: engle1990discussion167 reference: 'Engle, R. F. (1990), Discussion: Stock volatility and the crash of ''87, Review of Financial Studies 3(1), 103–106.'168 parents:169 - garch170 equation: $\sigma_t^2=\omega+\alpha(\varepsilon_{t-1}-\theta)^2+\beta\sigma_{t-1}^2$171 key_property: additive shift of the news-impact curve in return units (asymmetric GARCH)172 stationarity: $\alpha+\beta<1$; positivity $\omega>0$, $\alpha,\beta\ge0$173 status_in_wp46: out-of-scope174 reason: shift in return units is not scale-free; superseded by the NAGARCH form of Engle and Ng (1993), which is scored; no vetted implementation175- id: nagarch176 name: NAGARCH (Engle–Ng)177 short: NAGARCH178 year: 1993179 family: asymmetric180 bibkey: engle1993measuring181 parents:182 - garch183 - agarch184 equation: $\sigma_t^2=\omega+\alpha(\varepsilon_{t-1}-\theta\sigma_{t-1})^2+\beta\sigma_{t-1}^2$185 key_property: 'nonlinear asymmetric GARCH: news-impact curve shifted by $\theta\sigma_{t-1}$ (scale-free); sign/size bias tests'186 stationarity: persistence $\alpha(1+\theta^2)+\beta<1$187 status_in_wp46: scored188 registry_id: nagarch11189 note: Not the NGARCH of Higgins and Bera (1992) (power form); rugarch fGARCH has both as distinct submodels.190- id: vgarch191 name: VGARCH (Engle–Ng)192 short: VGARCH193 year: 1993194 family: asymmetric195 bibkey: engle1993measuring196 parents:197 - garch198 equation: $\sigma_t^2=\omega+\alpha(z_{t-1}+\theta)^2+\beta\sigma_{t-1}^2$, $z_{t-1}=\varepsilon_{t-1}/\sigma_{t-1}$199 key_property: news impact in standardized-shock units, independent of the variance level200 stationarity: '$\beta<1$ (the ARCH term is bounded in expectation: $\E(z+\theta)^2=1+\theta^2$)'201 status_in_wp46: out-of-scope202 reason: news impact does not scale with the variance level; dominated by NAGARCH in Engle and Ng's own comparison; no vetted implementation203- id: qgarch204 name: QGARCH (Sentana)205 short: QGARCH206 year: 1995207 family: asymmetric208 bibkey: sentana1995quadratic209 parents:210 - garch211 - agarch212 equation: $\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\psi\varepsilon_{t-1}+\beta\sigma_{t-1}^2$213 key_property: general quadratic form in lagged shocks; linear term gives a parabola-shifting asymmetry214 stationarity: $\alpha+\beta<1$; positivity iff $\psi^2\le4\alpha\omega$215 status_in_wp46: scored216 registry_id: qgarch11217 note: AGARCH is the special case psi = -2 alpha theta with the constant absorbed; the general QARCH has cross terms.218- id: avgarch219 name: AVGARCH / Taylor–Schwert220 short: AVGARCH221 year: 1986222 family: power223 bibkey: taylor2008modelling224 parents:225 - garch226 equation: $\sigma_t=\omega+\alpha|\varepsilon_{t-1}|+\beta\sigma_{t-1}$227 key_property: dynamics in the conditional standard deviation (absolute-value GARCH)228 stationarity: $\alpha\E|z|+\beta<1$ for the mean of $\sigma_t$; second moment needs $\E(\alpha|z|+\beta)^2<1$229 status_in_wp46: scored230 registry_id: avgarch11231 note: Taylor (1986, Wiley, 1st ed.; cited in the 2008 2nd ed., the only verified record) and Schwert (1989, schwert1989why).232- id: narch233 name: NARCH / NGARCH (Higgins–Bera)234 short: NARCH (Higgins–Bera)235 year: 1992236 family: power237 bibkey: higgins1992class238 parents:239 - arch240 - avgarch241 equation: $\sigma_t^{\delta}=\omega+\sum_{i}\alpha_i|\varepsilon_{t-i}|^{\delta}+\sum_{j}\beta_j\sigma_{t-j}^{\delta}$ (power-GARCH form; Higgins–Bera's original is the Box–Cox ARCH case)242 key_property: estimated power transformation of the ARCH recursion (nonlinear ARCH)243 stationarity: $\delta>0$; moment conditions of the APARCH type (ling2002necessary) with $\gamma=0$244 status_in_wp46: out-of-scope245 reason: nested in APARCH with gamma = 0 (scored); rugarch fGARCH(NGARCH) only; the name collides with NAGARCH246- id: aparch247 name: APARCH248 short: APARCH249 year: 1993250 family: power251 bibkey: ding1993long252 parents:253 - garch254 - avgarch255 - gjr256 - tgarch_zakoian257 - narch258 equation: $\sigma_t^{\delta}=\omega+\sum_{i}\alpha_i(|\varepsilon_{t-i}|-\gamma_i\varepsilon_{t-i})^{\delta}+\sum_{j}\beta_j\sigma_{t-j}^{\delta}$259 key_property: estimated power $\delta$ and asymmetry $\gamma$; nests GARCH, AVGARCH, GJR, TGARCH (Zakoïan), NGARCH260 stationarity: $\delta>0$, $|\gamma_i|<1$; $\sum_i\alpha_i\E(|z|-\gamma_iz)^{\delta}+\sum_j\beta_j<1$ (ding1993long; moments in ling2002necessary)261 status_in_wp46: scored262 registry_id: aparch11263 note: Also called PGARCH (EViews) or power GARCH; APARCH is the Ding–Granger–Engle name.264- id: hentschel_family265 name: Family GARCH (Hentschel)266 short: Family GARCH267 year: 1995268 family: power269 bibkey: hentschel1995all270 reference: 'Hentschel, L. (1995), All in the family: Nesting symmetric and asymmetric GARCH models, Journal of Financial Economics 39(1), 71–104.'271 parents:272 - aparch273 - egarch274 - nagarch275 - gjr276 - tgarch_zakoian277 - avgarch278 equation: $\dfrac{\sigma_t^{\lambda}-1}{\lambda}=\omega+\alpha\sigma_{t-1}^{\lambda}f^{\nu}(z_{t-1})+\beta\dfrac{\sigma_{t-1}^{\lambda}-1}{\lambda}$, $f(z)=|z-b|-c(z-b)$279 key_property: 'Box–Cox transform of $\sigma_t$ with a shifted and rotated absolute-value news-impact function: one nest for the asymmetric family'280 stationarity: family-specific (each nested member's condition); for $\lambda=\nu$ the recursion is a random-coefficient AR(1) in $\sigma_t^{\lambda}$, strictly stationary iff $\E\log(\lambda\alpha f^{\lambda}(z)+\beta)<0$281 status_in_wp46: out-of-scope282 reason: 'an encompassing nest, not a forecasting model: each nested member is scored separately; the free (lambda, nu) case is weakly identified; rugarch fGARCH is this family'283- id: duan_augmented284 name: Augmented GARCH (Duan)285 short: Augmented GARCH286 year: 1997287 family: power288 bibkey: duan1997augmented289 reference: Duan, J.-C. (1997), Augmented GARCH(p,q) process and its diffusion limit, Journal of Econometrics 79(1), 97–127.290 parents:291 - hentschel_family292 - aparch293 - egarch294 equation: $\phi_t=\omega+\xi_{1,t-1}\phi_{t-1}+\xi_{2,t-1}$, $\sigma_t^2=|\lambda\phi_t-\lambda+1|^{1/\lambda}$ ($\lambda\ne0$), $\xi_{1,t}=\alpha_1+\alpha_2|z_t-c|^{\delta}+\alpha_3\max(0,c-z_t)^{\delta}$,295 $\xi_{2,t}=\alpha_4\dfrac{|z_t-c|^{\delta}-1}{\delta}+\alpha_5\dfrac{\max(0,c-z_t)^{\delta}-1}{\delta}$ ((1,1) case)296 key_property: Box–Cox transform of the variance driven by Box–Cox functions of the shock; nests GARCH, EGARCH, GJR, TGARCH, APARCH, NAGARCH, VGARCH and the Hentschel family; diffusion limit297 stationarity: strict stationarity via $\E\log\xi_{1,t}<0$ (random-coefficient AR(1) in $\phi_t$)298 status_in_wp46: out-of-scope299 reason: encompassing nest (eight free shape parameters); no implementation in arch/rugarch; members scored individually300- id: figarch301 name: FIGARCH302 short: FIGARCH303 year: 1996304 family: fractional305 bibkey: baillie1996fractionally306 parents:307 - garch308 - igarch309 equation: $\sigma_t^2=\omega+\big[1-\beta L-(1-\phi L)(1-L)^{d}\big]\varepsilon_t^2+\beta\sigma_{t-1}^2$ (ARCH($\infty$) weights truncated in practice)310 key_property: 'fractional differencing: hyperbolic decay of shocks to the variance between GARCH ($d=0$) and IGARCH ($d=1$)'311 stationarity: strictly but not covariance stationary for $0<d\le1$ (no unconditional variance); positivity conditions in conrad2006inequality312 status_in_wp46: scored313 registry_id: figarch11314 note: arch truncates at 1,000 lags; the Chung (1999) re-parameterization is unverified and not cited.315- id: fiegarch316 name: FIEGARCH317 short: FIEGARCH318 year: 1996319 family: fractional320 bibkey: bollerslev1996modeling321 parents:322 - egarch323 - figarch324 equation: $\log\sigma_t^2=\omega+\phi(L)^{-1}(1-L)^{-d}\,[1+\psi(L)]\,g(z_{t-1})$, $g(z)=\theta z+\gamma(|z|-\E|z|)$325 key_property: fractional integration in the log variance with EGARCH asymmetry326 stationarity: covariance stationary (in $\log\sigma_t^2$) iff $d<1/2$; invertible for $d>-1/2$327 status_in_wp46: extension328 registry_id: fiegarch11329 reason: no vetted implementation in arch 8.0.0 / rugarch 1.5-6; custom implementation under way330- id: fiaparch331 name: FIAPARCH332 short: FIAPARCH333 year: 1998334 family: fractional335 bibkey: tse1998conditional336 parents:337 - aparch338 - figarch339 equation: $\sigma_t^{\delta}=\omega+\big\{1-[1-\beta(L)]^{-1}\phi(L)(1-L)^{d}\big\}(|\varepsilon_t|-\gamma\varepsilon_t)^{\delta}$340 key_property: fractional persistence combined with the APARCH power and asymmetry341 stationarity: as FIGARCH for $d$ (strict, not covariance stationary); $\delta>0$, $|\gamma|<1$342 status_in_wp46: extension343 registry_id: fiaparch11344 reason: no vetted implementation; custom implementation under way345- id: hygarch346 name: HYGARCH347 short: HYGARCH348 year: 2004349 family: fractional350 bibkey: davidson2004moment351 parents:352 - garch353 - figarch354 equation: $\sigma_t^2=\omega+\big\{1-[1-\beta(L)]^{-1}\phi(L)\big[1+\tau\big((1-L)^{d}-1\big)\big]\big\}\varepsilon_t^2$355 key_property: 'amplitude parameter $\tau$ interpolating GARCH ($\tau=0$) and FIGARCH ($\tau=1$): hyperbolic GARCH'356 stationarity: covariance stationary iff $\tau<1$ (Davidson 2004); positivity conditions non-trivial357 status_in_wp46: extension358 registry_id: hygarch11359 reason: no vetted implementation; custom implementation under way360- id: cgarch361 name: Component GARCH (Engle–Lee)362 short: CGARCH363 year: 1999364 family: component365 bibkey: engle1999long366 parents:367 - garch368 equation: $\sigma_t^2=q_t+\alpha(\varepsilon_{t-1}^2-q_{t-1})+\beta(\sigma_{t-1}^2-q_{t-1})$, $q_t=\omega+\rho q_{t-1}+\phi(\varepsilon_{t-1}^2-\sigma_{t-1}^2)$369 key_property: permanent (slowly mean-reverting) and transitory variance components370 stationarity: $0<\alpha+\beta<\rho<1$, $0<\phi<\min(\alpha,\beta)$; a GARCH(2,2) in reduced form371 status_in_wp46: scored372 registry_id: cgarch11373 note: Circulated as Engle and Lee (1993), 'A permanent and transitory component model...' (UCSD WP); rugarch csGARCH uses an equivalent parameterization.374- id: acgarch375 name: Asymmetric component GARCH376 short: Asym. CGARCH377 year: 1999378 family: component379 bibkey: engle1999long380 parents:381 - cgarch382 - gjr383 equation: $\sigma_t^2-q_t=\alpha(\varepsilon_{t-1}^2-q_{t-1})+\gamma(\varepsilon_{t-1}^2-q_{t-1})\ind{\varepsilon_{t-1}<0}+\beta(\sigma_{t-1}^2-q_{t-1})$, $q_t$ as in CGARCH384 key_property: leverage (GJR-type indicator) in the transitory component385 stationarity: as CGARCH with $\alpha+\gamma\kappa+\beta<\rho<1$386 status_in_wp46: out-of-scope387 reason: component and leverage effects are each scored separately (CGARCH, GJR); no vetted implementation of the combination (rugarch csGARCH has no leverage term)388- id: ding_granger_lm389 name: Long-memory ARCH by aggregation (Ding–Granger)390 short: LM-ARCH (Ding–Granger)391 year: 1996392 family: component393 bibkey: ding1996modeling394 parents:395 - ewma396 - garch397 equation: $\sigma_t^2=\sum_{k=1}^{N}w_k\sigma_{k,t}^2$, $\sigma_{k,t}^2=(1-\alpha_k)\varepsilon_{t-1}^2+\alpha_k\sigma_{k,t-1}^2$, $\sum_k w_k=1$398 key_property: aggregation of EWMA-type components with heterogeneous decay rates produces hyperbolic autocorrelations of $\varepsilon_t^2$399 stationarity: each component integrated; the $N$-component model with Beta-distributed $\alpha_k$ has long memory as $N\to\infty$400 status_in_wp46: out-of-scope401 reason: motivational construction; its fixed-weight practical descendant RM2006 is scored and its estimated two-component descendant is the CGARCH402- id: spline_garch403 name: Spline-GARCH404 short: Spline-GARCH405 year: 2008406 family: component407 bibkey: engle2008spline408 parents:409 - garch410 equation: $\sigma_t^2=\tau_t g_t$, $g_t=(1-\alpha-\beta)+\alpha\dfrac{\varepsilon_{t-1}^2}{\tau_{t-1}}+\beta g_{t-1}$, $\tau_t=c\exp\!\big(w_0t+\sum_{k=1}^{K}w_k\,((t-t_{k-1})_+)^2\big)$411 key_property: multiplicative low-frequency component as a deterministic exponential quadratic spline of time412 stationarity: $g_t$ unit-mean GARCH ($\alpha+\beta<1$); $\tau_t$ deterministic, so the process is not stationary413 status_in_wp46: out-of-scope414 reason: the spline is an in-sample fit of calendar time with no extrapolation rule for out-of-sample forecasts; not in arch/rugarch415- id: garch_midas416 name: GARCH-MIDAS417 short: GARCH-MIDAS418 year: 2013419 family: component420 bibkey: engle2013stock421 parents:422 - spline_garch423 - cgarch424 equation: $\sigma_t^2=\tau_t g_t$, $g_t=(1-\alpha-\beta)+\alpha\dfrac{\varepsilon_{t-1}^2}{\tau_t}+\beta g_{t-1}$, $\log\tau_t=m+\theta\sum_{k=1}^{K}\varphi_k(w)X_{t-k}$ (MIDAS weights on a low-frequency425 series $X$)426 key_property: long-run component driven by lagged low-frequency realized variance or macroeconomic variables through a MIDAS polynomial427 stationarity: $\alpha+\beta<1$ for $g_t$; $\tau_t$ inherits the properties of $X$428 status_in_wp46: extension429 registry_id: garch_midas430 reason: needs a low-frequency explanatory series; being implemented with the realized-variance driver (Tier B information set)431 note: arch's MIDASHyperbolic is a MIDAS-weighted ARCH, not GARCH-MIDAS.432- id: mc_garch_intraday433 name: Multiplicative component GARCH (intraday)434 short: MC-GARCH (intraday)435 year: 2011436 family: component437 bibkey: engle2011forecasting438 parents:439 - garch440 - spline_garch441 equation: $\sigma_{t,i}^2=h_t\,s_i\,q_{t,i}$, $q_{t,i}=\omega+\alpha\dfrac{\varepsilon_{t,i-1}^2}{h_t s_{i-1}}+\beta q_{t,i-1}$ (day $t$, intraday bin $i$)442 key_property: daily component $h_t$ $\times$ diurnal pattern $s_i$ $\times$ intraday GARCH $q_{t,i}$443 stationarity: $\alpha+\beta<1$ for $q_{t,i}$; $h_t$ from a daily model444 status_in_wp46: out-of-scope445 reason: intraday variance targets; WP46 forecasts daily and multi-day variances (rugarch mcsGARCH)446- id: swarch447 name: SWARCH (Hamilton–Susmel)448 short: SWARCH449 year: 1994450 family: regime451 bibkey: hamilton1994autoregressive452 parents:453 - arch454 equation: $\varepsilon_t=\sqrt{g_{S_t}}\,u_t$, $u_t=\sigma_t z_t$, $\sigma_t^2=\omega+\sum_{i=1}^{q}\alpha_i u_{t-i}^2$ ($+\xi\,u_{t-1}^2\ind{u_{t-1}<0}$), $S_t$ a $K$-state Markov chain455 key_property: Markov-switching scale factor on an ARCH process (ARCH only, to avoid path dependence)456 stationarity: ARCH condition within regimes; regime scales $g_k$ with $g_1=1$457 status_in_wp46: out-of-scope458 reason: ARCH-only components (no GARCH term); superseded by the Haas–Mittnik–Paolella MS-GARCH, which is scored459- id: ms_arch_cai460 name: MS-ARCH (Cai)461 short: MS-ARCH (Cai)462 year: 1994463 family: regime464 bibkey: cai1994markov465 parents:466 - arch467 - swarch468 equation: $\sigma_t^2=\omega_{S_t}+\sum_{i=1}^{q}\alpha_i\varepsilon_{t-i}^2$, $S_t\in\{1,2\}$ Markov (switching in the intercept)469 key_property: regime-switching ARCH intercept (switching in the level of the variance)470 stationarity: ARCH condition; regimes in the intercept only471 status_in_wp46: out-of-scope472 reason: ARCH-only; superseded by MS-GARCH (Haas et al.), which is scored473- id: ms_garch_gray474 name: MS-GARCH (Gray, collapsing)475 short: MS-GARCH (Gray)476 year: 1996477 family: regime478 bibkey: gray1996modeling479 parents:480 - garch481 - swarch482 equation: $\sigma_t^2=\omega_{S_t}+\alpha_{S_t}\varepsilon_{t-1}^2+\beta_{S_t}\tilde\sigma_{t-1}^2$, $\tilde\sigma_{t-1}^2=\sum_k p_{k,t-1}(\mu_k^2+\sigma_{k,t-1}^2)-\big(\sum_k p_{k,t-1}\mu_k\big)^2$483 key_property: 'first tractable MS-GARCH: the lagged variance is collapsed across regimes with the regime probabilities (removes path dependence)'484 stationarity: no simple closed-form condition; depends on the collapsing rule (regime-wise $\alpha_k+\beta_k<1$ is not the relevant condition)485 status_in_wp46: out-of-scope486 reason: collapsing approximation with a different likelihood from the HMP form; not implemented in MSGARCH or in WP46487- id: ms_garch_klaassen488 name: MS-GARCH (Klaassen)489 short: MS-GARCH (Klaassen)490 year: 2002491 family: regime492 bibkey: klaassen2002improving493 parents:494 - ms_garch_gray495 equation: $\sigma_t^2=\omega_{S_t}+\alpha_{S_t}\varepsilon_{t-1}^2+\beta_{S_t}\,\E\!\big[\sigma_{S_{t-1},t-1}^2\,\big|\,\F_{t-1},S_t\big]$496 key_property: collapses the lagged variance conditional on the current regime; analytic multi-step forecasts497 stationarity: as Gray's model, with the modified conditioning498 status_in_wp46: out-of-scope499 reason: collapsing variant; not the HMP definition used by WP46 and MSGARCH500- id: ms_garch_haas501 name: MS-GARCH (Haas–Mittnik–Paolella)502 short: MS-GARCH (HMP)503 year: 2004504 family: regime505 bibkey: haas2004new506 parents:507 - garch508 - ms_garch_gray509 equation: $\sigma_{k,t}^2=\omega_k+\alpha_k\varepsilon_{t-1}^2+\beta_k\sigma_{k,t-1}^2$ for each regime $k$; $r_t\,|\,S_t=k\sim D(\mu_k,\sigma_{k,t}^2)$; $S_t$ Markov with $P=(p_{ij})$510 key_property: $K$ parallel GARCH recursions driven by the common shock; analytic likelihood (Hamilton filter), stationarity conditions and forecasts511 stationarity: covariance stationary iff the spectral radius of a matrix built from $P$ and $\{\alpha_k,\beta_k\}$ is below one (haas2004new); regime-wise $\alpha_k+\beta_k<1$ is not necessary512 status_in_wp46: scored513 registry_id: msgarch2514 note: The definition implemented in the MSGARCH R package (ardia2019markov).515- id: ms_garch_pathdep516 name: Path-dependent MS-GARCH517 short: path-dep. MS-GARCH518 year: 2010519 family: regime520 bibkey: bauwens2010theory521 parents:522 - garch523 - ms_garch_gray524 equation: $\sigma_t^2=\omega_{S_t}+\alpha_{S_t}\varepsilon_{t-1}^2+\beta_{S_t}\sigma_{t-1}^2$, with $\sigma_{t-1}^2$ depending on the whole regime path $\{S_{t-1},S_{t-2},\dots\}$525 key_property: the literal regime-switching GARCH; likelihood integrates over $K^{t}$ paths (Bayesian MCMC, later EM/particle methods)526 stationarity: geometric ergodicity conditions in bauwens2010theory527 status_in_wp46: out-of-scope528 reason: simulation-based likelihood incompatible with the rolling re-estimation cadence of the benchmark (ML by augustyniak2014maximum)529- id: st_arch_hagerud530 name: Smooth-transition ARCH (Hagerud)531 short: ST-ARCH532 year: 1997533 family: regime534 bibkey: hagerud1997smooth535 parents:536 - arch537 - gjr538 equation: $\sigma_t^2=\omega+\sum_{i=1}^{q}\big[\alpha_i+\gamma_iF(\varepsilon_{t-i})\big]\varepsilon_{t-i}^2$, $F$ logistic (LSTARCH) or exponential (ESTARCH)539 key_property: smooth logistic/exponential transition between two ARCH regimes indexed by the lagged shock540 stationarity: $\sum_i(\alpha_i+\gamma_i\sup F)<1$ (sufficient)541 status_in_wp46: out-of-scope542 reason: ARCH-only precursor; the GARCH form of González-Rivera (1998) is the extension being implemented543- id: st_garch544 name: Smooth-transition GARCH (González-Rivera)545 short: ST-GARCH546 year: 1998547 family: regime548 bibkey: gonzalezrivera1998smooth549 parents:550 - garch551 - gjr552 - st_arch_hagerud553 equation: $\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\beta\sigma_{t-1}^2+(\delta_0+\delta_1\varepsilon_{t-1}^2)\,F(\varepsilon_{t-1})$, $F(x)=[1+\exp(-\theta x)]^{-1}$554 key_property: 'smooth (logistic) version of the GJR indicator: the speed $\theta$ of the transition is estimated; GJR is the limit $\theta\to\infty$'555 stationarity: 'sufficient: $\alpha+\max(0,\delta_1)+\beta<1$; positivity needs $\omega+\min(0,\delta_0)>0$'556 status_in_wp46: extension557 registry_id: stgarch558 reason: no vetted library implementation; custom implementation under way559- id: gas560 name: GAS / score-driven volatility561 short: GAS562 year: 2013563 family: score-driven564 bibkey: creal2013generalized565 parents:566 - garch567 equation: '$f_t=\omega+\alpha s_{t-1}+\beta f_{t-1}$, $s_t=S_t\,\partial\log p(r_t\,|\,f_t)/\partial f_t$; Student-$t$, $f_t=\sigma_t^2$: $s_t=\dfrac{\nu+3}{\nu}\Big[\dfrac{(\nu+1)\varepsilon_t^2}{(\nu-2)+\varepsilon_t^2/f_t}-f_t\Big]$'568 key_property: 'the update is the scaled score of the conditional density: GARCH for the normal, outlier-robust (Beta-$t$-GARCH) for the Student-$t$'569 stationarity: $|\beta|<1$ (the score is a martingale difference, $\E s_t=0$)570 status_in_wp46: scored571 registry_id: gas_t11572 note: Labelled a non-GARCH comparator; GAS-t(1,1) with inverse-information scaling coincides with Harvey's Beta-t-GARCH up to scaling.573- id: beta_t_egarch574 name: Beta-$t$-EGARCH (DCS)575 short: Beta-t-EGARCH576 year: 2008577 family: score-driven578 bibkey: harvey2008beta579 parents:580 - egarch581 - gas582 equation: $\log\sigma_t^2=\omega+\phi\log\sigma_{t-1}^2+\kappa u_{t-1}$, $u_t=\dfrac{(\nu+1)\varepsilon_t^2}{\nu\sigma_t^2+\varepsilon_t^2}-1$ ($u_t+1$ Beta-distributed)583 key_property: log variance driven by the bounded Student-$t$ score; closed-form moments and multi-step forecasts584 stationarity: $|\phi|<1$; $u_t$ i.i.d. with $\E u_t=0$585 status_in_wp46: extension586 reason: custom implementation under way (Harvey 2013 book harvey2013dynamic; leverage/skewness version harvey2014egarch)587- id: garch_x588 name: GARCH-X (realized regressor)589 short: GARCH-X590 year: 2002591 family: realized592 bibkey: engle2002new593 parents:594 - garch595 equation: $\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\gamma x_{t-1}+\beta\sigma_{t-1}^2$596 key_property: lagged realized measure enters the GARCH variance additively597 stationarity: $\alpha+\beta<1$, $\gamma\ge0$; the long-run level depends on the dynamics of $x_t$598 status_in_wp46: scored599 registry_id: garchx_rv11600- id: mem601 name: MEM (multiplicative error model)602 short: MEM603 year: 2002604 family: realized605 bibkey: engle2002new606 parents:607 - garch608 equation: $x_t=\mu_t\eta_t$, $\mu_t=\omega+\alpha x_{t-1}+\beta\mu_{t-1}$, $\eta_t\ge0$ i.i.d., $\E\eta_t=1$609 key_property: GARCH-type dynamics for a positive-valued series (realized variance, range, duration); ACD is the duration case610 stationarity: $\alpha+\beta<1$611 status_in_wp46: out-of-scope612 reason: 'not a model of r_t; its realized-measure equation is the second equation of the scored HEAVY model (multiple-indicator version: engle2006multiple; ACD: Engle and Russell 1998)'613- id: heavy614 name: HEAVY615 short: HEAVY616 year: 2010617 family: realized618 bibkey: shephard2010realising619 parents:620 - garch_x621 - mem622 equation: $\sigma_t^2=\omega+\alpha x_{t-1}+\beta\sigma_{t-1}^2$; $\mu_t=\E[x_t\,|\,\F_{t-1}]=\omega_R+\alpha_Rx_{t-1}+\beta_R\mu_{t-1}$623 key_property: the realized measure replaces the squared return in the return-variance equation; a second MEM equation forecasts the measure624 stationarity: $\beta<1$ and $\alpha_R+\beta_R<1$625 status_in_wp46: scored626 registry_id: heavy11627- id: realized_garch628 name: Realized GARCH629 short: Realized GARCH630 year: 2012631 family: realized632 bibkey: hansen2012realized633 parents:634 - garch_x635 - egarch636 equation: $\log\sigma_t^2=\omega+\beta\log\sigma_{t-1}^2+\gamma\log x_{t-1}$; $\log x_t=\xi+\varphi\log\sigma_t^2+\tau_1z_t+\tau_2(z_t^2-1)+u_t$637 key_property: joint model of returns and the realized measure with a measurement equation and a leverage function $\tau(z)$638 stationarity: $|\beta+\varphi\gamma|<1$639 status_in_wp46: scored640 registry_id: realgarch11641 note: Log-linear form scored; the likelihood includes the measurement equation and is not comparable with GARCH likelihoods.642- id: realized_egarch643 name: Realized EGARCH644 short: Realized EGARCH645 year: 2016646 family: realized647 bibkey: hansen2016exponential648 parents:649 - realized_garch650 - egarch651 equation: $\log\sigma_t^2=\omega+\beta\log\sigma_{t-1}^2+\tau(z_{t-1})+\gamma u_{t-1}$; $\log x_t=\xi+\varphi\log\sigma_t^2+\delta(z_t)+u_t$652 key_property: EGARCH-type leverage from the return shock plus the measurement innovation $u_t$ in the variance; several realized measures at once653 stationarity: $|\beta|<1$654 status_in_wp46: extension655 registry_id: realegarch11656 reason: custom implementation under way657- id: har658 name: HAR-RV659 short: HAR-RV660 year: 2009661 family: realized662 bibkey: corsi2009simple663 parents: []664 equation: $x_{t+1}=\beta_0+\beta_d x_t+\beta_w\bar x_t^{(5)}+\beta_m\bar x_t^{(22)}+e_{t+1}$, $\bar x_t^{(n)}=n^{-1}\sum_{k=0}^{n-1}x_{t-k}$665 key_property: heterogeneous autoregression of realized variance on daily, weekly and monthly averages (approximate long memory by OLS)666 stationarity: AR(22) with restricted coefficients; stationary iff its characteristic roots lie outside the unit circle667 status_in_wp46: scored668 registry_id: har_rv669 note: Levels and log (smearing-corrected) versions scored, one direct regression per horizon; working paper 2004.670- id: harq671 name: HARQ (and HAR-CJ, HAR-RS)672 short: HARQ673 year: 2016674 family: realized675 bibkey: bollerslev2016exploiting676 parents:677 - har678 equation: $x_{t+1}=\beta_0+(\beta_d+\beta_q\,RQ_t^{1/2})x_t+\beta_w\bar x_t^{(5)}+\beta_m\bar x_t^{(22)}+e_{t+1}$ ($RQ_t$ = realized quarticity)679 key_property: measurement-error-adjusted persistence; HAR-CJ splits continuous and jump variation, HAR-RS uses realized semivariances680 note: 'HAR-CJ: andersen2007roughing; HAR-RS: patton2015good; HARQ: bollerslev2016exploiting (all in references.bib).'681 stationarity: as HAR with time-varying daily coefficient682 status_in_wp46: out-of-scope683 reason: needs realized quarticity / bipower variation / semivariances as additional inputs; the pre-registered realized tier uses the plain HAR684- id: vec685 name: VEC-GARCH (Bollerslev–Engle–Wooldridge)686 short: VEC687 year: 1988688 family: multivariate689 bibkey: bollerslev1988capital690 reference: Bollerslev, T., Engle, R. F. and Wooldridge, J. M. (1988), A capital asset pricing model with time-varying covariances, Journal of Political Economy 96(1), 116–131.691 parents:692 - garch693 equation: $\mathrm{vech}(H_t)=C+\sum_{i}A_i\,\mathrm{vech}(\varepsilon_{t-i}\varepsilon_{t-i}')+\sum_{j}B_j\,\mathrm{vech}(H_{t-j})$ (diagonal $A_i,B_j$ in the original)694 key_property: 'first multivariate GARCH: every covariance is a linear function of lagged cross-products and covariances'695 stationarity: eigenvalues of $\sum A_i+\sum B_j$ inside the unit circle; positive definiteness not guaranteed696 status_in_wp46: out-of-scope697 reason: O(N^4) parameters and no positive-definiteness guarantee; the BEKK and DCC descendants are scored698- id: factor_arch699 name: Factor-ARCH (Engle–Ng–Rothschild)700 short: Factor-ARCH701 year: 1990702 family: multivariate703 bibkey: engle1990asset704 reference: 'Engle, R. F., Ng, V. K. and Rothschild, M. (1990), Asset pricing with a factor-ARCH covariance structure: Empirical estimates for Treasury bills, Journal of Econometrics 45(1–2), 213–237.'705 parents:706 - vec707 - garch708 equation: $H_t=\Omega+\sum_{k=1}^{K}\beta_k\beta_k'\,\sigma_{k,t}^2$, $\sigma_{k,t}^2$ a univariate GARCH of the factor portfolio $\phi_k'r_t$709 key_property: time-varying covariances through a few GARCH factors (reduced rank); positive definite if $\Omega$ is710 stationarity: factor GARCH conditions711 status_in_wp46: out-of-scope712 reason: requires factor-portfolio choice; Tier C baskets have N <= 5 where full correlation models are feasible713- id: ccc714 name: CCC-GARCH715 short: CCC716 year: 1990717 family: multivariate718 bibkey: bollerslev1990modelling719 parents:720 - garch721 equation: $H_t=D_tRD_t$, $D_t=\mathrm{diag}(\sigma_{1,t},\dots,\sigma_{N,t})$, $R$ constant722 key_property: univariate GARCH margins with a constant correlation matrix723 stationarity: univariate conditions; $H_t$ PD iff $R$ PD724 status_in_wp46: scored725 registry_id: ccc_garch11726- id: bekk727 name: BEKK (full / diagonal)728 short: BEKK729 year: 1995730 family: multivariate731 bibkey: engle1995multivariate732 parents:733 - vec734 - factor_arch735 equation: $H_t=CC'+\sum_{i}A_i'\varepsilon_{t-i}\varepsilon_{t-i}'A_i+\sum_{j}B_j'H_{t-j}B_j$736 key_property: quadratic form guaranteeing positive definiteness; nests the diagonal VEC and factor-ARCH737 stationarity: eigenvalues of $\sum_i(A_i\otimes A_i)'+\sum_j(B_j\otimes B_j)'$ inside the unit circle738 status_in_wp46: out-of-scope739 reason: O(N^2) parameters per matrix; the scalar restriction with covariance targeting is the scored Tier C member740- id: bekk_scalar741 name: Scalar BEKK (covariance targeting)742 short: scalar BEKK743 year: 1995744 family: multivariate745 bibkey: engle1995multivariate746 parents:747 - bekk748 - ewma749 equation: $H_t=(1-a-b)\bar\Sigma+a\,\varepsilon_{t-1}\varepsilon_{t-1}'+b\,H_{t-1}$750 key_property: two dynamic parameters; EWMA covariance is the case $a+b=1$, $\bar\Sigma$-term dropped751 stationarity: $a,b\ge0$, $a+b<1$752 status_in_wp46: scored753 registry_id: scalar_bekk11754- id: dcc755 name: DCC756 short: DCC757 year: 2002758 family: multivariate759 bibkey: engle2002dynamic760 parents:761 - ccc762 - bekk_scalar763 equation: $H_t=D_tR_tD_t$; $Q_t=(1-a-b)\bar Q+a\,z_{t-1}z_{t-1}'+b\,Q_{t-1}$; $R_t=\mathrm{diag}(Q_t)^{-1/2}Q_t\,\mathrm{diag}(Q_t)^{-1/2}$764 key_property: scalar GARCH-type dynamics for the correlation of the standardized residuals; two-step estimation765 stationarity: $a,b\ge0$, $a+b<1$; the two-step estimator with $\bar Q$ = sample correlation is not consistent in general (aielli2013dynamic)766 status_in_wp46: scored767 registry_id: dcc11768 note: 'Theoretical companion: engle2001theoretical (NBER WP 8554).'769- id: adcc770 name: Asymmetric DCC771 short: aDCC772 year: 2006773 family: multivariate774 bibkey: cappiello2006asymmetric775 parents:776 - dcc777 - gjr778 equation: $Q_t=(1-a-b)\bar Q-g\bar N+a\,z_{t-1}z_{t-1}'+g\,n_{t-1}n_{t-1}'+b\,Q_{t-1}$, $n_t=z_t\odot\ind{z_t<0}$779 key_property: correlations respond more to joint negative shocks (GJR logic in the correlation recursion)780 stationarity: $a+b+g\,\delta<1$ with $\delta$ the largest eigenvalue of $\bar Q^{-1/2}\bar N\bar Q^{-1/2}$781 status_in_wp46: scored782 registry_id: adcc11783- id: cdcc784 name: Corrected DCC (cDCC)785 short: cDCC786 year: 2013787 family: multivariate788 bibkey: aielli2013dynamic789 parents:790 - dcc791 equation: $Q_t=(1-a-b)\bar S+a\,Q_{t-1}^{*1/2}z_{t-1}z_{t-1}'Q_{t-1}^{*1/2}+b\,Q_{t-1}$, $Q_t^{*}=\mathrm{diag}(Q_t)$792 key_property: 'rescales the outer product so that $\bar S$ is the unconditional mean of $Q_t$: consistent two-step estimation'793 stationarity: as DCC794 status_in_wp46: out-of-scope795 reason: the scored DCC uses Engle's original recursion with correlation targeting; cDCC (rmgarch default) is not implemented796- id: go_garch797 name: GO-GARCH798 short: GO-GARCH799 year: 2002800 family: multivariate801 bibkey: vanderweide2002gogarch802 parents:803 - factor_arch804 - ccc805 equation: $\varepsilon_t=Zf_t$, $H_t=ZV_tZ'$, $V_t=\mathrm{diag}(\sigma_{1,t}^2,\dots,\sigma_{N,t}^2)$ with univariate GARCH for each factor; $Z=P\Lambda^{1/2}U$, $U$ orthogonal806 key_property: generalized orthogonal factors (an estimated rotation of the principal components) with independent univariate GARCH807 stationarity: univariate factor conditions808 status_in_wp46: out-of-scope809 reason: identification of the rotation (ML/ICA) is implementation-specific; not part of the pre-registered Tier C set810 note: 'Precursor: Alexander''s O-GARCH (principal components, U = I; not verified, not cited).'811- id: deco812 name: DECO (dynamic equicorrelation)813 short: DECO814 year: 2012815 family: multivariate816 bibkey: engle2012dynamic817 reference: Engle, R. F. and Kelly, B. (2012), Dynamic equicorrelation, Journal of Business & Economic Statistics 30(2), 212–228.818 parents:819 - dcc820 equation: $R_t=(1-\rho_t)I_N+\rho_t\iota\iota'$, $\rho_t=\dfrac{1}{N(N-1)}\big(\iota'R_t^{\mathrm{DCC}}\iota-N\big)$821 key_property: all pairwise correlations equal at each date; closed-form inverse and determinant for large $N$822 stationarity: as the underlying DCC; $-1/(N-1)<\rho_t<1$823 status_in_wp46: out-of-scope824 reason: designed for large N; Tier C baskets have N <= 5825- id: dcc_nl826 name: DCC-NL (nonlinear shrinkage)827 short: DCC-NL828 year: 2019829 family: multivariate830 bibkey: engle2019large831 parents:832 - dcc833 - cdcc834 equation: DCC/cDCC recursion with $\bar Q$ replaced by the nonlinear-shrinkage estimator of the unconditional correlation matrix and composite-likelihood estimation of $(a,b)$835 key_property: 'large-$N$ DCC: nonlinear shrinkage of the targeting matrix and composite-likelihood estimation'836 note: 'Composite likelihood for vast DCC: pakel2021fitting.'837 stationarity: as DCC838 status_in_wp46: out-of-scope839 reason: large-N estimator; unnecessary for N <= 5840- id: mv_heavy841 name: Multivariate HEAVY842 short: MV-HEAVY843 year: 2012844 family: multivariate845 bibkey: noureldin2012multivariate846 parents:847 - heavy848 - bekk_scalar849 equation: $H_t=C_H+a_H\,RC_{t-1}+b_H\,H_{t-1}$; $M_t=\E[RC_t\,|\,\F_{t-1}]=C_M+a_M\,RC_{t-1}+b_M\,M_{t-1}$ (scalar form; $RC_t$ = realized covariance)850 key_property: realized covariance drives both the return covariance and its own (Wishart-type) measurement equation851 stationarity: $b_H<1$, $a_M+b_M<1$852 status_in_wp46: out-of-scope853 reason: needs synchronized intraday data for realized covariances (hansen2014realized, archakov2026multivariate are the realized-GARCH analogues); Tier C is daily-only854- id: mv_realized_garch855 name: Multivariate Realized GARCH856 short: MV Realized GARCH857 year: 2026858 family: multivariate859 bibkey: archakov2026multivariate860 parents:861 - realized_garch862 - mv_heavy863 equation: $v_t=\omega+\beta v_{t-1}+\gamma u_{t-1}$ with $v_t=(\log h_{1,t},\dots,\log h_{N,t},\mathrm{vecl}[\log C_t]')'$ and realized-measure equation $x_t=\xi+\varphi v_t+\tau(z_t)+u_t$ (schematic;864 see reference)865 key_property: realized-GARCH structure on the log variances and the matrix-logarithm of the correlation matrix (unconstrained parametrization)866 stationarity: $|\mathrm{eig}(\beta+\gamma\varphi)|<1$ (vector form)867 status_in_wp46: out-of-scope868 reason: needs realized covariances; 2026 Journal of Econometrics article, outside the pre-registered Tier C set869- id: sv870 name: Stochastic volatility (log-SV, Taylor)871 short: SV (Taylor)872 year: 1986873 family: stochastic-volatility874 bibkey: taylor2008modelling875 parents: []876 equation: $\varepsilon_t=\sigma_tz_t$, $\log\sigma_t^2=\omega+\phi\log\sigma_{t-1}^2+\sigma_\eta\eta_t$, $\eta_t\sim N(0,1)$ independent of $z_t$877 key_property: latent (parameter-driven) log variance with its own innovation; the variance is not $\F_{t-1}$-measurable878 stationarity: $|\phi|<1$879 status_in_wp46: unsupported880 registry_id: sv881 reason: simulation-based (MCMC/particle) likelihood not feasible at the rolling re-estimation cadence; registry reference kim1998stochastic (MCMC with mixture offset) and taylor1994modeling (review);882 Bayesian analysis in jacquier1994bayesian883 note: Taylor (1986, Wiley 1st ed.); the verified record is the 2008 2nd edition.884- id: msv885 name: Multivariate SV, QML (Harvey–Ruiz–Shephard)886 short: MSV (Harvey–Ruiz–Shephard)887 year: 1994888 family: stochastic-volatility889 bibkey: harvey1994multivariate890 parents:891 - sv892 equation: $\log\varepsilon_{i,t}^2=\log\sigma_{i,t}^2+\log z_{i,t}^2$, $\log\sigma_t^2=\omega+\Phi\log\sigma_{t-1}^2+\eta_t$ (linear state space; Kalman-filter QML)893 key_property: multivariate SV with common factors; quasi-maximum likelihood on $\log\varepsilon^2$894 stationarity: eigenvalues of $\Phi$ inside the unit circle895 status_in_wp46: out-of-scope896 reason: parameter-driven multivariate model; outside the GARCH-based Tier C set897- id: sv_ksc898 name: SV by MCMC (Kim–Shephard–Chib)899 short: SV (Kim–Shephard–Chib)900 year: 1998901 family: stochastic-volatility902 bibkey: kim1998stochastic903 parents:904 - sv905 equation: log-SV as above; $\log z_t^2$ approximated by a seven-component normal mixture; MCMC with an offset mixture and particle-filter likelihood for model comparison with ARCH906 key_property: exact-likelihood (Bayesian) inference for SV and a formal comparison with GARCH907 stationarity: as log-SV908 status_in_wp46: unsupported909 reason: same architecture as sv; estimation method only (registry reference for the unsupported SV entry)910- id: periodic_garch911 name: Periodic GARCH912 short: Periodic GARCH913 year: 1996914 family: other915 bibkey: bollerslev1996periodic916 reference: Bollerslev, T. and Ghysels, E. (1996), Periodic autoregressive conditional heteroscedasticity, Journal of Business & Economic Statistics 14(2), 139–151.917 parents:918 - garch919 equation: $\sigma_t^2=\omega_{s(t)}+\alpha_{s(t)}\varepsilon_{t-1}^2+\beta_{s(t)}\sigma_{t-1}^2$, $s(t)\in\{1,\dots,S\}$ the stage of a known periodic cycle920 key_property: GARCH coefficients that vary deterministically with the calendar (day of the week, intraday season)921 stationarity: 'periodic stationarity: $\prod_{s=1}^{S}(\alpha_s+\beta_s)<1$ (second order, symmetric case)'922 status_in_wp46: out-of-scope923 reason: calendar effects are not part of the pre-registered information set; no implementation in arch/rugarch924- id: garch_jump925 name: GARCH-jump (ARJI, Maheu–McCurdy)926 short: GARCH-jump (ARJI)927 year: 2004928 family: other929 bibkey: maheu2004news930 reference: Maheu, J. M. and McCurdy, T. H. (2004), News arrival, jump dynamics, and volatility components for individual stock returns, Journal of Finance 59(2), 755–793.931 parents:932 - garch933 - gjr934 equation: $r_t=\mu+\sigma_tz_t+\sum_{k=1}^{n_t}Y_{t,k}$, $n_t\,|\,\F_{t-1}\sim\mathrm{Poisson}(\lambda_t)$, $\lambda_t=\lambda_0+\rho\lambda_{t-1}+\gamma\xi_{t-1}$, $\sigma_t^2$ GARCH/GJR on $\varepsilon_{t-1}=r_{t-1}-\mu$935 key_property: autoregressive conditional jump intensity on top of a GARCH diffusion component; total variance $\sigma_t^2+\lambda_t(\mu_J^2+\sigma_J^2)$936 stationarity: $\rho<1$ for $\lambda_t$; GARCH condition for $\sigma_t^2$937 status_in_wp46: out-of-scope938 reason: jump-intensity filter requires the full compound-Poisson likelihood; variance forecasts mix two components; no vetted implementation939 note: 'Earlier ARCH-jump mixture: Jorion (1988, RFS 1(4)) (not verified, not cited).'940