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\title{\color{report_main}{Assignment Econometrics 2024}}		% Title
\author{Hendrik Marcel W Tillemans}						% Author
\date{\today}											% Date

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\begin{document}

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\begin{titlepage}
	\centering
    \vspace*{0.5 cm}
    \includegraphics[scale = 0.95]{../figures/vub.png}\\[1.0 cm]	% University Logo
    \textsc{\LARGE \newline\newline Free University Brussels}\\[2.0 cm]	% University Name
	\textsc{\Large \color{report_main}{Class: Econometrics}}\\[0.5 cm]				% Course Code
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	{ \huge \bfseries \thetitle}\\
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	\begin{minipage}{0.5\textwidth}
		\begin{flushleft} \large
			\emph{Professor:}\\
			Jeroen Kerkhof\\
            Faculty of Economic Sciences\\
			\end{flushleft}
			\end{minipage}~
			\begin{minipage}{0.4\textwidth}
            
			\begin{flushright} \large
			\emph{Group:} \\
Hendrik Marcel W Tillemans\\
             
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	\end{minipage}\\[2 cm]

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\section{Question 1}

This my answer to question 1.

\subsection{Example}
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\pagebreak

creates a page break.

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\section{Simulation Study}

\subsection{1.2: Linear Fit on Generated Data}

\begin{table}[h]
\input{table_1_2}
\caption{Linear Fit on Generated Data}
\label{tab::table_1_2}
\end{table}
 
\subsection{1.3: Linear Fit with 1 Variable}

\begin{table}[h]
\input{table_1_3}
\caption{Linear Fit with 1 Variable}
\label{tab::table_1_3}
\end{table}

\subsection{1.4: New Linear Fit on Generated Data}

\begin{table}[h]
\input{table_1_4}
\caption{New Linear Fit on Generated Data}
\label{tab::table_1_4}
\end{table}

\subsection{1.5: New Linear Fit with 1 Variable}

\begin{table}[h]
\input{table_1_5}
\caption{Linear Fit with 1 Variable}
\label{tab::table_1_5}
\end{table}

\subsection{1.6: Generate Data with Small Variance on x1}

\begin{table}[h]
\input{table_1_6}
\caption{Generate Data with Small Variance on x1}
\label{tab::table_1_6}
\end{table}

\section{examples}
Some greek letters:

$\alpha$
$\beta$
$\gamma$
$\theta$
$\varepsilon$
$\pi$
$\lambda$
$\tau$

$x=x+27$
x=x+27



$A \Longrightarrow B$


$\underbrace{abs}_{test}$

sub and superscript

$\beta_0$
$\sum_{i=1}^{n} i$

In an equation:

\begin{equation}
\sum_{j=1}{n} j^2 \beta
\end{equation}

Equation without number

\begin{equation*}
A \Rightarrow B
\end{equation*}


\section{Empirical Investigation}

Here is some example code to create tables and graphs from the 
Python script. In order for this to work you would first need 
to run the script non\_linear\_models\_example\_report.py. Running that 
file (using the recommended file structure) creates some figures 
in the figures folder and some tables in .tex files in the report folder.


\subsection{Question 3}

For instance, here the file df\_table.tex is used print the actual numbers 
in the table.

\begin{table}[h]
\input{df_table}
\caption{This tables has the estimates for ...}
\label{tab::estimation_results}
\end{table}

\subsection{Question 4: Some graphs}

\begin{figure}
\includegraphics[width=0.6\paperwidth]{../figures/quadratic_model_y}
\caption{This is a Figure coming straight from Python.}
\label{fig::example_data}
\end{figure}

In Figure \ref{fig::example_data} we see the data.

\begin{figure}
\includegraphics[width=0.6\paperwidth]{../figures/quadratic_model_linear}
\caption{This is a linear fit on a quadratic model.}
\label{fig::example_quadratic_linear}
\end{figure}

In Figure \ref{fig::example_quadratic_linear} we see a linear fit.


\begin{figure}
\includegraphics[width=0.6\paperwidth]{../figures/quadratic_model_quadratic}
\caption{This is quadratic fit on a quadratic model.}
\label{fig::example_quadratic_quadratic}
\end{figure}

In Figure \ref{fig::example_quadratic_quadratic} we see that

\subsection{Question 5}

Equation example with matrices:

\begin{equation}\label{eq::wald_test}
H_0: \beta_1 = - \beta_2; \beta_3=0; \beta_2 + 2\beta_4 = 2
\quad H_1: \neg H_0
\end{equation}

can be written in matrix form as:
\begin{equation}\label{eq::matrix_form}
\begin{bmatrix}
1 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 2
\end{bmatrix}
\begin{bmatrix}
\beta_1 \\
\beta_2 \\
\beta_3 \\
\beta_4
\end{bmatrix} = 
\begin{bmatrix}
0 \\
0 \\
2
\end{bmatrix}
\end{equation}

In equation \eqref{eq::wald_test} we see that... and in equation \eqref{eq::matrix_form} we see that

\subsection{Question 6}


$\beta$

\begin{table}
\input{summary}
\caption{This tables has the estimates summary}
\label{tab::estimation_results_summary}
\end{table}

Table \ref{tab::estimation_results_summary} has the full summary.

\begin{table}
\input{results_coef}
\caption{This tables has the estimates summary}
\label{tab::estimation_results_coef}
\end{table}

Table \ref{tab::estimation_results_coef} has the only the coefficient 
results.


\end{document}
