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\documentclass{article}
\usepackage{doc,url,verbatim,fancyvrb}
\usepackage{pifont}
\usepackage[authoryear]{natbib}
\usepackage[pdftex]{graphicx}
\usepackage{gretl}
\usepackage[letterpaper,body={6.3in,9.15in},top=.8in,left=1.1in]{geometry}
%\usepackage[pdftex,hyperfootnotes=false]{hyperref}
%\usepackage[a4paper,body={6.1in,9.7in},top=.8in,left=1.1in]{geometry}
\begin{document}
\setlength{\parindent}{0pt}
\setlength{\parskip}{1ex}
\newcommand{\argname}[1]{\textsl{#1}}
\title{ivpanel version 1.0}
\author{Allin Cottrell}
\date{September 22, 2020}
\maketitle
\section{Introduction}
This package estimates three variants of panel data models with
instrumental variables, namely fixed-effects, the ``between'' model,
and random effects (Generalized 2-Stage Least Squares or G2SLS). An
extended discussion of these models can be found in chapter 7 of
\cite{baltagi05}. The sample script provided with this package
replicates results obtained by Baltagi using \textsf{Stata} for
each of the three supported models.
The required arguments to the main public function \texttt{ivpanel()}
are as follows:
\begin{enumerate}
\item \argname{y} (series), the dependent variable
\item \argname{X} (list), the regressors, both exogenous and endogenous
\item \argname{Z} (list), the instruments, including any exogenous
regressors
\end{enumerate}
Note that exogenous variables in list \argname{X} should be repeated
in \argname{Z}, as in gretl's \texttt{tsls} command. The constant is
automatically added to \argname{X} and \argname{Z} if it's not already
present.
An optional fourth argument, \argname{case}, can be used to specify
the type of model. The argument is an integer switch that takes values
1, 2 or 3 (this may be extended in future). A value of 1 means fixed
effects, and is implicit if no fourth argument is given; a value of 2
means to use the between estimator; and a value of 3 means G2SLS.
Leaving aside the constant, all the members of \argname{X} and
\argname{Z} must be time-varying when the fixed effects case is
selected.
An optional final boolean argument, \argname{quiet}, controls the
printing of output: if \argname{quiet} is set to a non-zero value,
printing is suppressed.
\section{The ivpanel bundle}
By way of return value, \texttt{ivpanel()} offers a gretl bundle
containing the items shown in Table~\ref{tab:bun}.
\begin{table}[htbp]
\centering
\begin{tabular}{llp{.6\textwidth}}
\textit{name} & \textit{type} & \textit{description} \\[4pt]
\texttt{case} & scalar & the value of the \argname{case} argument on input \\
\texttt{nobs} & scalar & the total number of observations used \\
\texttt{coeff} & matrix & column vector of coefficients \\
\texttt{stderr} & matrix & column vector of standard errors \\
\texttt{vcv} & matrix & the coefficient covariance matrix \\
\texttt{uhat} & matrix & the vector of residuals \\
\texttt{yhat} & matrix & the vector of fitted values \\
\texttt{SSR} & scalar & sum of squared residuals \\
\texttt{sigma} & scalar & standard error of the regression \\
\texttt{df} & scalar & degrees of the freedom for the regression \\
\texttt{rsq} & scalar & correlation-based $R^2$ \\
\texttt{wald} & scalar & Wald joint $\chi^2$ test for all regressors \\
\texttt{modstr} & string & short description of the model \\
\texttt{ystr} & string & the name of the dependent variable \\
\texttt{Estr} & string & the names of the endogenous regressor(s) \\
\texttt{Istr} & string & the names of the (additional) instruments \\
\texttt{dims} & matrix & data dimensions (cases 1 and 3 only) \\
\texttt{Fpool} & scalar & $F$-test for joint significance of fixed
effects (case 1 only)
\end{tabular}
\caption{Items in ivpanel bundle}
\label{tab:bun}
\end{table}
Some comments on the bundle members follow.
\begin{itemize}
\item The $R^2$ value, \texttt{rsq}, is calculated as the square of the
correlation between the dependent variable and the fitted values.
\item The \texttt{dims} vector, if present, contains three elements
holding, respectively, the number of groups used and the minimum and
maximum time-series spans.
\item The \texttt{Fpool} statistic for fixed effects is calculated as
per \cite{wooldridge90}, adjusted for the panel case. The formula is
\[
\frac{\mbox{SSRr} - \mbox{SSRu}}{\mbox{SSRf}} \times
\frac{\mbox{dfd}}{\mbox{dfn}}
\]
where SSRf is the sum of squared residuals from the fixed-effects IV
model and the other two SSR values are calculated thus:
\begin{enumerate}
\item We run the fixed-effects first-stage regressions and save the
fitted values; we then use these to replace the endogenous
regressors.
\item SSRr is then obtained via OLS, and SSRu from a fixed-effects
regression. SSRu differs from SSRf in that it uses the ``raw''
residuals, without correction as one usually applies with two stage
least squares (i.e.\ replacing the first-stage fitted values with
the actual data for the endogenous regressors when computing the
fitted values).
\end{enumerate}
\end{itemize}
\section{Sample script}
The sample script estimates three models of the (log) crime rate
across the counties of North Carolina over the years 1981 to 1987,
using data from \cite{cornwell94}. The endogenous regressors are
\texttt{lpolpc} (log of police per capita) and \texttt{lprbarr} (log
of the estimated probability of arrest). The instruments (besides the
exogenous regressors) are \texttt{ltaxpc} (log of tax revenue per
capita), \texttt{lmix} (log of the offense ``mix'', face-to-face
versus other) and a set of year dummies.
It's perhaps noteworthy that despite the IV approach, the coefficient
on \texttt{lpolpc} is positive in all the models, the opposite of what
would be expected if the estimator had succeeded in picking up a
causal relationship. At least in the fixed effects specification the
positive \texttt{lpolpc} coefficient is not statistically significant
($P$-value > 0.4).
\section{Graphical interface}
An entry-point for \textsf{ivpanel} can be found under the
\textsf{Panel} sub-menu of gretl's \textsf{Model} menu: the label is
``Panel IV model.'' See Figure~\ref{fig:gui}. The specification of the
endogenous regressors is implicit: any variables that appear in the
``regressors'' list and not in the ``instruments'' list are taken to
be endogenous.
\begin{figure}[htbp]
\centering
\includegraphics[scale=0.6]{ivpanel-gui}
\caption{Specify arguments for ivpanel}
\label{fig:gui}
\end{figure}
\section{Auxiliary printing function}
The auxiliary public function \texttt{ivp\_print()} is provided to
``pretty-print'' the results contained in the bundle provided by
\texttt{ivpanel()}; \texttt{ivp\_print()} takes a pointer to the
bundle as its sole argument.
\bibliographystyle{gretl}
\bibliography{gretl}
\end{document}
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