% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/onatski_2010.R
\name{.onatski_2010}
\alias{.onatski_2010}
\title{Onatski (2010) Edge Distribution Estimator}
\usage{
.onatski_2010(eigenvalues, kmax, n_iter = 4L)
}
\arguments{
\item{eigenvalues}{Numeric vector of eigenvalues in descending order,
typically obtained from \code{.extract_eigenvalues()}. Must
contain at least \code{kmax + 5} elements to allow the OLS
regression in the calibration step.}

\item{kmax}{Integer. Maximum number of factors to consider.}

\item{n_iter}{Integer. Maximum number of iterations for the
calibration procedure. Defaults to \code{4} as recommended
by Onatski (2010).}
}
\value{
A named list with the following elements:
\describe{
\item{k}{Integer. Estimated number of factors.}
\item{delta}{Numeric. The estimated threshold \eqn{\delta = 2|\hat{\beta}|}.}
\item{beta}{Numeric. The estimated slope coefficient \eqn{\hat{\beta}}
from the OLS regression in the final iteration.}
\item{differences}{Numeric vector of length kmax. Successive
eigenvalue differences \eqn{\lambda_i - \lambda_{i+1}}.}
\item{n_iter}{Integer. Number of iterations performed.}
}
}
\description{
Estimates the number of factors using the Edge Distribution (ED)
estimator of Onatski (2010). The estimator exploits the fact that
idiosyncratic eigenvalues of the sample covariance matrix cluster
around a single point, while systematic eigenvalues diverge to
infinity. The threshold separating the two groups is estimated
iteratively using the square root shape of the edge of the
eigenvalue distribution.
}
\details{
The ED estimator of Onatski (2010) is based on the theoretical result
that idiosyncratic eigenvalues cluster around the upper edge
\eqn{u(\mathcal{F}^{c,A,B})} of the limiting spectral distribution,
while systematic eigenvalues diverge. Near the edge, the density of
the limiting spectral distribution behaves like a square root function,
implying that eigenvalue differences \eqn{\lambda_i - \lambda_{i+1}}
for idiosyncratic eigenvalues behave approximately as
\eqn{(an)^{-2/3}}.

The calibration procedure estimates \eqn{\hat{\beta} = (an)^{-2/3}}
by regressing five consecutive eigenvalues \eqn{\lambda_j, \ldots,
  \lambda_{j+4}} on a constant and \eqn{(j-1)^{2/3}, \ldots,
  (j+3)^{2/3}}, where \eqn{j} is initialized at \eqn{r_{max} + 1}
and updated iteratively.

The estimator requires \code{eigenvalues} to contain at least
\code{kmax + 5} elements so that the OLS window \eqn{j, \ldots, j+4}
is always available.
}
\references{
Onatski, A. (2010). Determining the Number of Factors From Empirical
Distribution of Eigenvalues. \emph{The Review of Economics and
Statistics}, 92(4), 1004-1016.
}
\seealso{
\code{.extract_eigenvalues()}, \code{\link{select_factors}}
}
