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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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	\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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		\begin{flushleft} \large
			\emph{Professor:}\\
			Jeroen Kerkhof\\
            Faculty of Economic Sciences\\
			\end{flushleft}
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			\emph{Group:} \\
Hendrik Marcel W Tillemans\\
             
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\section{Simulation Study}


\subsection{Question 1.2}
 Are the estimates of $\beta_0$, $\beta_1$ and $\beta_2$ close to their true values? Why (not)?
 
 
We investigate a linear model with noise

\[y=\beta_0 + \beta_1 x1 + \beta_2 x2 + u\]

where

\[x1 \sim \mathcal{N}(3,\,6)\]
\[x2 \sim \mathcal{N}(3,\,6)\]
\[u \sim \mathcal{N}(0,\,3)\]

In figure \ref{fig::plot_1_1} we have a 3D representation of the generated model.

\begin{figure}[hb]
\includegraphics[width=0.6\paperwidth]{../figures/question_1_1}
\caption{Generated points for Question 1.1.}
\label{fig::plot_1_1}
\end{figure}



\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{Question 1.3}
Compare your estimates with those of question 1.2. Wich model do you choose? Discuss in terms of $\beta_1$ and model prediction.

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

\subsection{Question 1.4}
Do the results confirm what you would have expected to change in your estimation results compared to the results in question 1.2? Why (not)? How about the standard errors of the estimates of $\beta_1$ and $\beta_2$?
\begin{table}[h]
\input{table_1_4}
\caption{New Linear Fit on Generated Data}
\label{tab::table_1_4}
\end{table}

\subsection{Question 1.5}
Are the OLS estimators for the slope coefficients biased? Why (not)?
\begin{table}[h]
\input{table_1_5}
\caption{Linear Fit with 1 Variable}
\label{tab::table_1_5}
\end{table}

\subsection{Question 1.6}
Do the results confirm what you would have
expected to change in your estimation results compared to the results in question 1.2?
Why (not)? How about the standard errors of the estimates of $\beta_1$ ? Use the formula
Var$\beta_1$ to motivate your answer. What would happen if the standard deviation of x1
is equal to 0 instead of equals 1? Discuss in terms of the assumptions of the Multiple
Linear Regression mode.
\begin{table}[h]
\input{table_1_6}
\caption{Generate Data with Small Variance on x1}
\label{tab::table_1_6}
\end{table}

\begin{figure}[hb]
\includegraphics[width=0.6\paperwidth]{../figures/question_1_6}
\caption{Generated points for Question 1.6.}
\label{fig::plot_1_6}
\end{figure}

\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}
