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#' no external instruments are available or to supplement external instruments to improve the efficiency of the

#' IV estimator.

#' Consider the model in the equation:

#' \ifelse{html}{\out{<br><div style="text-align:center">y<sub>t</sub>=&beta;<sub>0</sub>+&beta;<sub>1</sub>P<sub>t</sub>+&beta;<sub>2</sub>X<sub>t</sub>+&epsilon;<sub>t</sub></div>}}{\deqn{ y_{t}=\beta_{0}+ \beta_{1} P_{t} + \beta_{2} X_{t} + \epsilon_{t}}}

#'

#' where \eqn{t=1,..,T} indexes either time or cross-sectional units.The endogeneity problem arises from the correlation of

#' @details

#'

#' Let's consider the model:

#' \ifelse{html}{\out{<br><div style="text-align:center">Y<sub>t</sub>=&beta;<sub>0</sub>+&alpha;P<sub>t</sub>+&epsilon;<sub>t</sub></div>}}{ \deqn{Y_{t} = \beta_{0} + \alpha P_{t} + \epsilon_{t}}}

#' \ifelse{html}{\out{<div style="text-align:center">P<sub>t</sub>=&pi;'Z<sub>t</sub>+&nu;<sub>t</sub></div>}}{ \deqn{P_{t}=\pi^{'}Z_{t} + \nu_{t}}}

#'

#' \code{latentIV} considers \ifelse{html}{\out{Z<sub>t</sub>'}}{\eqn{Z_{t}^{'}}} to be a latent, discrete, exogenous variable with an unknown number of groups \eqn{m} and \eqn{\pi} is a vector of group means.

#' It is assumed that \eqn{Z} is independent of the error terms \eqn{\epsilon} and \eqn{\nu} and that it has at least two groups with different means.

#' The structural and random errors are considered normally distributed with mean zero and variance-covariance matrix \eqn{\Sigma}:

#' \ifelse{html}{\out{<div style="text-align:center">&Sigma;=(&sigma;<sub>&epsilon;</sub><sup>2</sup>, &sigma;<sub>0</sub><sup>2</sup>,

#' <br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&sigma;<sub>0</sub><sup>2</sup>, &sigma;<sub>&nu;</sub><sup>2</sup>)</div>}}{

#' \deqn{\Sigma = \left(

#' \subsection{Method}{

#'

#' Consider the model:

#' \ifelse{html}{\out{<br><div style="text-align:center">Y<sub>t</sub>=&beta;<sub>0</sub> + &beta;<sub>1</sub>X<sub>t</sub>+&alpha;P<sub>t</sub>+&epsilon;<sub>t</sub></div>}}{\deqn{ Y_{t} = \beta_{0}+ \beta_{1}X_{t} + \alpha P_{t}+\epsilon_{t} \hspace{0.3cm} (1) }}

#'

#' \ifelse{html}{\out{<div style="text-align:center">P<sub>t</sub>=Z<sub>t</sub>+&nu;<sub>t</sub></div>}}{\deqn{ P_{t} = \gamma Z_{t}+\nu_{t} \hspace{2.5 cm} (2)}}

#'

#'

#' Consider the model:

#' \ifelse{html}{\out{<br><div style="text-align:center">Y<sub>t</sub>=&beta;<sub>0</sub>+&beta;<sub>1</sub>P<sub>t</sub>+&beta;<sub>2</sub>X<sub>t</sub>+&epsilon;<sub>t</sub></div>}}{\deqn{Y_{t}=\beta_{0}+ \beta_{1} P_{t} + \beta_{2} X_{t} + \epsilon_{t}}}

#'

#' where \eqn{t=1,..,T} indexes either time or cross-sectional units, \ifelse{html}{\out{Y<sub>t</sub>}}{\eqn{Y_{t}}} is a \eqn{1x1} response variable,

#'

#' The marginal distribution of the endogenous regressor \ifelse{html}{\out{P<sub>t</sub>}}{\eqn{P_{t}}} is obtained using the Epanechnikov

#' kernel density estimator (Epanechnikov, 1969), as below:

#' \ifelse{html}{\out{<br><div style="text-align:center">h&#770;(p)=1/(T&#183;b) &sum;(K&#183;((p-P<sub>t</sub>)/b))</div>}}{\deqn{\hat{h}(p)=\frac{1}{T\cdot b}\sum_{t=1}^{T}K\left(\frac{p-P_{t}}{b}\right)}}

#'

#' where \ifelse{html}{\out{P<sub>t</sub>}}{\eqn{P_{t}}} is the endogenous regressor,