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\title{Limited mobility bias correction}

\author{Simen Gaure}
\address{Ragnar Frisch Centre for Economic Research, Oslo, Norway}
\date{November 12, 2015}

\begin{document}
\begin{abstract}
A description of what limited bias correction is. And how to compensate for it.
\end{abstract}
\maketitle



\section{Introduction}
We assume we have an OLS model in matrix form
\begin{equation}\label{osys}
 Y = X\beta + D\alpha + F\gamma + \epsilon
\end{equation}
where \(X\) is a \((n \times k)\)-matrix, \(D\) is a \((n \times
g_1)\)-matrix, and \(F\) is a \((n\times g_2)\)-matrix.  It is assumed
that \(D\) and \(F\) are matrices of dummy variables and that both
\(g_1\) and \(g_2\) are very large (as in \(10^5-10^7\)). The
\pkg{lfe}-package estimates \(\alpha\) and \(\gamma\) as OLS
coefficients.  Some applications study the variances \(\var(D\alpha)\)
and \(\var(F\gamma)\), as well as the covariance \(\cov(D\alpha,
F\gamma)\). It was shown in \cite{AGSU07} that if one uses the
estimates \(D\hat\alpha\) and \(F\hat\gamma\) for this purpose, the
resulting variances are positively biased, and the covariance is
typically negatively biased.  The biases can be large enough to change
the sign of the covariance. They also provided explicit formulas for
the size of the bias, in terms of the trace of some very large
matrices.

In short, the bias for \(\var(D\hat\alpha)\) can be computed as
\begin{equation}\label{bias}
  \delta_\alpha = \sigma_\epsilon^2 \tr(A),
\end{equation}
where \(A\) is a large matrix depending only on \(X\), \(D\) and \(F\).
The unbiased variance can then be estimated as
\begin{equation}\label{unbias}
  \var(D\alpha) = \var(D\hat\alpha) - \sigma_\epsilon^2\tr(A)
\end{equation}
In some typical instances, the matrix \(A\) will be too large to handle directly,
being of size \(g_1\).
The bias for \(\var(F\hat\gamma)\) and the covariance, are of the same form.

\section{How lfe handles this}
In \cite{GS14} a method for estimating \(\delta_\alpha\) was outlined.
It uses the fact that 
\begin{equation}
\tr(A) = \E(x^t A x),
\end{equation}
if \(x\) is a vector of independent random variables \(x_i\) with \(\E(x_i) = 0\) and
\(\var(x_i) = 1\).  In fact, each \(x_i\) can optimally be choosen as drawing
from \(\{-1,1\}\) with uniform probability.  This works because
even if \(A\) is too large to be handled directly, it is possible to compute
\(A x\) for a vector \(x\).  Thus, \pkg{lfe}'s \code{fevcov} routine uses
a sample mean to estimate \(\tr(A)\).

\section{Another method}
Formula \ref{unbias} suggests another method for estimating \(\var(D\alpha)\).
We rewrite it as
\begin{equation}
\var(D\hat\alpha) = \var(D\alpha) + \sigma_\epsilon^2\tr(A)
\end{equation}
Now, \(b=\var(D\alpha)\) and \(a=\tr(A)\) are constants, so 
we can write
\begin{equation}\label{sampleeq}
\var(D\hat\alpha) = a\sigma_\epsilon^2 + b
\end{equation}
If we resample and scale the residuals \(\hat\epsilon\), and do the estimation of
\(\hat\alpha\) over again, we will obtain a series of different observations
of formula \ref{sampleeq}.  We may then do an OLS on model \ref{sampleeq}, and
the estimate \(\hat a\) will be an estimate for \(\tr(A)\) which can be
used in equation \ref{unbias}.  The \(\var(D\hat\alpha)\) is not normally distributed
around its mean, this typically skews \(\hat b\) more than \(\hat a\), so \(\hat a\) is
a better estimate.  \pkg{lfe} does not implement this method.


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\bibitem{AGSU07}
M.~Andrews, L.~Gill, T.~Schank, and R.~Upward, \emph{{High wage workers and low
  wage firms: negative assortative matching or limited mobility bias?}},
  Journal of the Royal Statistical Society(A) \textbf{171(3)} (2008), 673--697.

\bibitem{GS14}
S.~Gaure, \emph{Correlation bias correction in two-way fixed-effects linear
  regression}, Stat \textbf{3} (2014), no.~1, 370--390.

\end{thebibliography}

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