% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/svp_information_criteria.R
\name{svp_IC}
\alias{svp_IC}
\title{Information criteria for SV(p) AR-order selection}
\usage{
svp_IC(
  fit,
  criteria = c("BIC_Kalman", "AIC_Kalman", "BIC_HR", "AIC_HR"),
  filter_method = c("mixture", "corrected", "particle"),
  proxy = c("bayes_optimal", "u"),
  K = 7L,
  M = 1000L,
  seed = 42L,
  del = 1e-10
)
}
\arguments{
\item{fit}{Output of \code{\link{svp}}. Must carry the original \code{y}
series (which \code{svp()} stores by default), \code{errorType}, and
\code{leverage} fields.}

\item{criteria}{Character vector. Subset of
\code{c("BIC_Kalman", "AIC_Kalman", "AICc_Kalman", "BIC_Whittle",
"BIC_HR", "AIC_HR", "BIC_YW", "AIC_YW")}. Default returns the four
recommended criteria: \code{c("BIC_Kalman", "AIC_Kalman", "BIC_HR",
"AIC_HR")}. See the description for the rationale for each
opt-in criterion.}

\item{filter_method}{Character. Filter method passed to
\code{\link{filter_svp}} for \code{*_Kalman} criteria. One of
\code{"mixture"} (default, recommended), \code{"corrected"}, or
\code{"particle"}.}

\item{proxy}{Character. Leverage proxy passed to \code{\link{filter_svp}}.
\code{"bayes_optimal"} (default here, unlike \code{filter_svp})
removes the \eqn{O(T)} log-likelihood bias of the û-proxy under
Student-t leverage and restores Schwarz consistency of \code{BIC_Kalman}.
\code{"u"} reproduces the paper-faithful (Remark 3.5) Kalman likelihood;
set this if you need IC values that match the filter's default behavior.}

\item{K}{Integer. Number of mixture components for
\code{filter_method = "mixture"}. Default 7 (KSC).}

\item{M}{Integer. Number of particles for
\code{filter_method = "particle"}. Default 1000. Ignored for other
filter methods.}

\item{seed}{Integer. Random seed for the bootstrap particle filter.
Default 42. Ignored for non-particle filters.}

\item{del}{Numeric. Small constant added inside \eqn{\log} to avoid
\eqn{\log 0}. Default \code{1e-10}.}
}
\value{
Named numeric vector of the requested criteria. Lower is better.
}
\description{
Computes information criteria for an \code{\link{svp}} fit to support
AR-order selection. Eight criteria are computable; \strong{four are
returned by default} — \code{BIC_Kalman}, \code{AIC_Kalman},
\code{BIC_HR}, \code{AIC_HR}. These span two families (state-space QML
and Hannan--Rissanen two-stage ARMA) and two penalty philosophies
(Schwarz-consistent BIC / Shibata-efficient AIC), and were selected as
the most informative criteria across the simulation grid of the
SVHT methodology paper (Ahsan, Dufour and Rodriguez-Rondon 2026).
The remaining four are available on request via the \code{criteria}
argument.
}
\details{
\strong{Default criteria (returned by \code{svp_IC(fit)}):}
\itemize{
  \item \code{BIC_Kalman}, \code{AIC_Kalman}: \eqn{-2\,\hat\ell_K + k\log T}
        and \eqn{-2\,\hat\ell_K + 2k} where \eqn{\hat\ell_K} is the
        (quasi-)log-likelihood from \code{\link{filter_svp}}; default
        \code{filter_method = "mixture"} uses the Gaussian mixture
        Kalman filter (Kim, Shephard and Chib 1998). \code{BIC_Kalman}
        is the primary recommended criterion: Schwarz-consistent under
        the Bayes-optimal leverage proxy (see \code{proxy} argument)
        and strong finite-sample performance across the simulation
        grid (Ahsan, Dufour and Rodriguez-Rondon 2026).
        \code{AIC_Kalman} is Shibata-efficient
        and often selects larger \eqn{p} sooner at \eqn{p_{\mathrm{true}}
        \ge 2}.
  \item \code{BIC_HR}, \code{AIC_HR}: Hannan--Rissanen (1982) two-stage
        ARMA(\eqn{p,p}) criteria. Stage 1: long-AR pre-whitening at
        order \eqn{L = \lfloor 1.5\, T^{1/3}\rfloor} produces residuals
        \eqn{\hat\varepsilon_t}. Stage 2: OLS regression of \eqn{y_t^*}
        on AR lags \eqn{1{:}p} of \eqn{y_t^*} and MA lags \eqn{1{:}p}
        of \eqn{\hat\varepsilon_t} gives \eqn{\hat\sigma_u^2}. Then
        \eqn{T_{\mathrm{eff}} \log \hat\sigma_u^2 + \{2(2p{+}1),
        (2p{+}1)\log T_{\mathrm{eff}}\}}. Filter-free anchor, robust to
        mis-specification of the GMKF mixture. \code{BIC_HR} is
        Schwarz-consistent for ARMA(\eqn{p,p}) (Hannan & Rissanen
        1982; Pötscher 1989).
}

\strong{Opt-in criteria (request via \code{criteria = ...}):}
\itemize{
  \item \code{AICc_Kalman}: \code{AIC_Kalman} with the Hurvich--Tsai
        (1989) small-sample correction \eqn{2k(k+1)/(T-k-1)}.
        Numerically equivalent to \code{AIC_Kalman} at \eqn{T \ge 500};
        use when \eqn{T < 500}.
  \item \code{BIC_Whittle}: \eqn{-2\,\hat\ell_W + k\log T} where
        \eqn{\hat\ell_W} is the Whittle log-likelihood evaluated at the
        SV(\eqn{p}) signal-plus-noise spectral density
        \eqn{f(\omega) = \sigma_v^2 / |1 - \sum_j \phi_j e^{-ij\omega}|^2
                       + \sigma_\varepsilon^2(\nu)}.
        Schwarz-consistent but collapses to \eqn{\hat p = 1} in 98--100\%
        of cells at \eqn{p_{\mathrm{true}} \ge 2} under near-unit-root
        persistence (Ahsan, Dufour and Rodriguez-Rondon 2026). Useful
        as a conservative
        diagnostic: a Whittle selection of \eqn{p > 1} is strong
        evidence against \eqn{p = 1}.
  \item \code{AIC_YW}, \code{BIC_YW}: \emph{Legacy / not recommended.}
        Yule--Walker projection-error criteria on
        \eqn{y_t^* = \log(y_t^2 + \delta) - \mu}, computed as
        \eqn{T \log \hat\sigma_{\mathrm{pred}}^2 + \{2k, k\log T\}}
        with the AR(\eqn{p}) projection-error variance. Under
        SV(\eqn{p}), \eqn{y_t^*} is ARMA(\eqn{p,p}) (not AR(\eqn{p})),
        so the AR projection error does not saturate at
        \eqn{p_{\mathrm{true}}} and the criteria are
        \emph{inconsistent}: the AR(\eqn{p}) projection-error variance
        keeps decreasing past \eqn{p_{\mathrm{true}}}, producing
        non-monotone (sometimes anti-Schwarz) behaviour in \eqn{T}.
        Simulation evidence: 0--29\% correct selection at
        \eqn{p_{\mathrm{true}} = 2} across all DGP cells and
        \eqn{T \le 10{,}000} (Ahsan, Dufour and Rodriguez-Rondon
        2026). Retained for paper-reproducibility of the
        documented failure-case results; \strong{use \code{BIC_HR} /
        \code{AIC_HR} for theoretically consistent AR-order selection.}
}

Lower is better; \code{argmin} over a grid of candidate \code{p} (see
\code{\link{svp_AR_order}}) selects the AR order.
}
\section{Leverage invariance of non-Kalman criteria}{

Leverage does not affect \code{AIC_YW}, \code{BIC_YW}, or
\code{BIC_Whittle}: under the W-ARMA-SV parameterization
\eqn{\mathrm{Cov}(v_t, \varepsilon_{t-1}) = 0} for all three error
distributions (odd-times-even moment symmetry), so the autocovariance
structure of \eqn{y_t^*} is invariant to the leverage parameter. The
\code{*_HR} and \code{*_Kalman} criteria do incorporate leverage
through the estimated \eqn{\delta_p} and the conditional state
innovation variance.
}

\examples{
\donttest{
set.seed(1)
y <- sim_svp(2000, phi = 0.95, sigy = 1, sigv = 0.5)$y
fit1 <- svp(y, p = 1)
fit2 <- svp(y, p = 2)
svp_IC(fit1)
svp_IC(fit2)
}

}
\references{
Akaike, H. (1974). A new look at the statistical model identification.
\emph{IEEE Transactions on Automatic Control} 19(6), 716--723.
\doi{10.1109/TAC.1974.1100705}

Schwarz, G. (1978). Estimating the dimension of a model.
\emph{Annals of Statistics} 6(2), 461--464.
\doi{10.1214/aos/1176344136}

Shibata, R. (1976). Selection of the order of an autoregressive model
by Akaike's information criterion. \emph{Biometrika} 63(1), 117--126.
\doi{10.1093/biomet/63.1.117}

Hannan, E. J. (1980). The estimation of the order of an ARMA process.
\emph{Annals of Statistics} 8(5), 1071--1081.
\doi{10.1214/aos/1176345144}

Hannan, E. J., and Rissanen, J. (1982). Recursive estimation of mixed
autoregressive-moving average order. \emph{Biometrika} 69(1), 81--94.
\doi{10.1093/biomet/69.1.81}

Pötscher, B. M. (1989). Model selection under nonstationarity:
Autoregressive models and stochastic linear regression models.
\emph{Annals of Statistics} 17(3), 1257--1274.
\doi{10.1214/aos/1176347267}

Whittle, P. (1953). Estimation and information in stationary time series.
\emph{Arkiv f\"or Matematik} 2, 423--434. \doi{10.1007/BF02590998}

Dunsmuir, W. (1979). A central limit theorem for parameter estimation in
stationary vector time series and its application to models for a signal
observed with noise. \emph{Annals of Statistics} 7(3), 490--506.
\doi{10.1214/aos/1176344671}

Hurvich, C. M., and Tsai, C.-L. (1989). Regression and time series model
selection in small samples. \emph{Biometrika} 76(2), 297--307.
\doi{10.1093/biomet/76.2.297}

Kim, S., Shephard, N., and Chib, S. (1998). Stochastic volatility:
likelihood inference and comparison with ARCH models.
\emph{Review of Economic Studies} 65(3), 361--393.
\doi{10.1111/1467-937X.00050}

White, H. (1982). Maximum likelihood estimation of misspecified models.
\emph{Econometrica} 50(1), 1--25. \doi{10.2307/1912526}

Ahsan, M. N., Dufour, J.-M., and Rodriguez-Rondon, G. (2026). Estimation and
inference for stochastic volatility models with heavy-tailed distributions.
Bank of Canada Staff Working Paper 2026-8. \doi{10.34989/swp-2026-8}
}
\seealso{
\code{\link{svp_AR_order}}, \code{\link{svp}}, \code{\link{filter_svp}}
}
