323 lines
7.4 KiB
TeX
323 lines
7.4 KiB
TeX
\documentclass[12pt]{article}
|
|
\usepackage{natbib}
|
|
\usepackage{url}
|
|
\usepackage[utf8x]{inputenc}
|
|
\usepackage{mathtools}%
|
|
\usepackage{graphicx}
|
|
\usepackage{parskip}
|
|
\usepackage{xcolor}%
|
|
\usepackage{fancyhdr}
|
|
\usepackage{vmargin}
|
|
\usepackage{booktabs}%
|
|
\usepackage{sectsty}% for coloring sections
|
|
\setmarginsrb{3 cm}{2.5 cm}{3 cm}{2.5 cm}{1 cm}{1.5 cm}{1 cm}{1.5 cm}
|
|
|
|
% define your own custom colors
|
|
% If you want to change the colors you would need to update the RGB code in the
|
|
% last brackets. Better not change the name of the color as it is used elsewhere
|
|
\definecolor{report_main}{HTML}{200045}
|
|
\definecolor{report_second}{HTML}{F39912}
|
|
\definecolor{report_third}{HTML}{8B0010}
|
|
|
|
\title{\color{report_main}{Assignment Econometrics 2024}} % Title
|
|
\author{Hendrik Marcel W Tillemans} % Author
|
|
\date{\today} % Date
|
|
|
|
\makeatletter
|
|
\let\thetitle\@title
|
|
\let\theauthor\@author
|
|
\let\thedate\@date
|
|
\makeatother
|
|
|
|
\pagestyle{fancy}
|
|
\fancyhf{}
|
|
\rhead{\theauthor} % header on the right
|
|
\lhead{\thetitle} % header on the left
|
|
\cfoot{\thepage} % footer in the center
|
|
|
|
\sectionfont{\color{report_main}}
|
|
\subsectionfont{\color{report_third}}
|
|
|
|
%% Add pagebreak before each section
|
|
\let\oldsection\section
|
|
\renewcommand\section{\clearpage\oldsection}
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% This is where the actual document starts
|
|
%
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\begin{document}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% This section details the group information
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\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
|
|
\rule{\linewidth}{0.2 mm} \\[0.4 cm]
|
|
{ \huge \bfseries \thetitle}\\
|
|
\rule{\linewidth}{0.2 mm} \\[1.5 cm]
|
|
|
|
\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\\
|
|
|
|
\end{flushright}
|
|
|
|
\end{minipage}\\[2 cm]
|
|
|
|
% takes the current date
|
|
\thedate
|
|
|
|
\end{titlepage}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% This details the inclusion (or not) of the table of contents
|
|
% and list of figures and tables.
|
|
% You can add/remove page breaks as you seem fit.
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\tableofcontents
|
|
|
|
\pagebreak
|
|
|
|
\listoffigures
|
|
|
|
\listoftables
|
|
|
|
\pagebreak
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% This is the start of the actual document content
|
|
% You can just write text in here as you would in any other word processor.
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
|
|
\section{Simulation Study}
|
|
|
|
\subsection{1.1: Generate Simulation Data}
|
|
|
|
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{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}
|
|
|
|
\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}
|