332 lines
8.2 KiB
TeX
332 lines
8.2 KiB
TeX
\documentclass[12pt]{article}
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\usepackage{natbib}
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\usepackage{url}
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\usepackage[utf8x]{inputenc}
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\usepackage{mathtools}%
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\usepackage{graphicx}
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\usepackage{parskip}
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\usepackage{xcolor}%
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\usepackage{fancyhdr}
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\usepackage{vmargin}
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\usepackage{booktabs}%
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\usepackage{sectsty}% for coloring sections
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\setmarginsrb{3 cm}{2.5 cm}{3 cm}{2.5 cm}{1 cm}{1.5 cm}{1 cm}{1.5 cm}
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% define your own custom colors
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% If you want to change the colors you would need to update the RGB code in the
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% last brackets. Better not change the name of the color as it is used elsewhere
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\definecolor{report_main}{HTML}{200045}
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\definecolor{report_second}{HTML}{F39912}
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\definecolor{report_third}{HTML}{8B0010}
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\title{\color{report_main}{Assignment Econometrics 2024}} % Title
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\author{Hendrik Marcel W Tillemans} % Author
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\date{\today} % Date
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\makeatletter
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\let\thetitle\@title
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\let\theauthor\@author
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\let\thedate\@date
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\makeatother
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\pagestyle{fancy}
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\fancyhf{}
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\rhead{\theauthor} % header on the right
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\lhead{\thetitle} % header on the left
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\cfoot{\thepage} % footer in the center
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\sectionfont{\color{report_main}}
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\subsectionfont{\color{report_third}}
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%% Add pagebreak before each section
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\let\oldsection\section
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\renewcommand\section{\clearpage\oldsection}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% This is where the actual document starts
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%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% This section details the group information
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{titlepage}
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\centering
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\vspace*{0.5 cm}
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\includegraphics[scale = 0.95]{../figures/vub.png}\\[1.0 cm] % University Logo
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\textsc{\LARGE \newline\newline Free University Brussels}\\[2.0 cm] % University Name
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\textsc{\Large \color{report_main}{Class: Econometrics}}\\[0.5 cm] % Course Code
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\rule{\linewidth}{0.2 mm} \\[0.4 cm]
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{ \huge \bfseries \thetitle}\\
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\rule{\linewidth}{0.2 mm} \\[1.5 cm]
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\begin{minipage}{0.5\textwidth}
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\begin{flushleft} \large
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\emph{Professor:}\\
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Jeroen Kerkhof\\
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Faculty of Economic Sciences\\
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\end{flushleft}
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\end{minipage}~
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\begin{minipage}{0.4\textwidth}
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\begin{flushright} \large
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\emph{Group:} \\
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Hendrik Marcel W Tillemans\\
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\end{flushright}
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\end{minipage}\\[2 cm]
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% takes the current date
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\thedate
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\end{titlepage}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% This details the inclusion (or not) of the table of contents
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% and list of figures and tables.
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% You can add/remove page breaks as you seem fit.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\tableofcontents
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\pagebreak
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\listoffigures
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\listoftables
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\pagebreak
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% This is the start of the actual document content
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% You can just write text in here as you would in any other word processor.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Simulation Study}
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\subsection{Question 1.2}
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Are the estimates of $\beta_0$, $\beta_1$ and $\beta_2$ close to their true values? Why (not)?
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We investigate a linear model with noise
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\[y=\beta_0 + \beta_1 x1 + \beta_2 x2 + u\]
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where
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\[x1 \sim \mathcal{N}(3,\,6)\]
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\[x2 \sim \mathcal{N}(3,\,6)\]
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\[u \sim \mathcal{N}(0,\,3)\]
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In figure \ref{fig::plot_1_1} we have a 3D representation of the generated model.
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\begin{figure}[hb]
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\includegraphics[width=0.6\paperwidth]{../figures/question_1_1}
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\caption{Generated points for Question 1.1.}
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\label{fig::plot_1_1}
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\end{figure}
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\subsection{1.2: Linear Fit on Generated Data}
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\begin{table}[h]
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\input{table_1_2}
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\caption{Linear Fit on Generated Data}
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\label{tab::table_1_2}
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\end{table}
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\subsection{Question 1.3}
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Compare your estimates with those of question 1.2. Wich model do you choose? Discuss in terms of $\beta_1$ and model prediction.
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\begin{table}[h]
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\input{table_1_3}
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\caption{Linear Fit with 1 Variable}
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\label{tab::table_1_3}
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\end{table}
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\subsection{Question 1.4}
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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 $\beta_2$?
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\begin{table}[h]
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\input{table_1_4}
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\caption{New Linear Fit on Generated Data}
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\label{tab::table_1_4}
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\end{table}
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\subsection{Question 1.5}
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Are the OLS estimators for the slope coefficients biased? Why (not)?
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\begin{table}[h]
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\input{table_1_5}
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\caption{Linear Fit with 1 Variable}
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\label{tab::table_1_5}
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\end{table}
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\subsection{Question 1.6}
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Do the results confirm what you would have
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expected to change in your estimation results compared to the results in question 1.2?
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Why (not)? How about the standard errors of the estimates of $\beta_1$ ? Use the formula
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Var$\beta_1$ to motivate your answer. What would happen if the standard deviation of x1
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is equal to 0 instead of equals 1? Discuss in terms of the assumptions of the Multiple
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Linear Regression mode.
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\begin{table}[h]
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\input{table_1_6}
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\caption{Generate Data with Small Variance on x1}
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\label{tab::table_1_6}
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\end{table}
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\begin{figure}[hb]
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\includegraphics[width=0.6\paperwidth]{../figures/question_1_6}
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\caption{Generated points for Question 1.6.}
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\label{fig::plot_1_6}
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\end{figure}
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\section{examples}
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Some greek letters:
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$\alpha$
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$\beta$
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$\gamma$
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$\theta$
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$\varepsilon$
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$\pi$
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$\lambda$
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$\tau$
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$x=x+27$
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x=x+27
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$A \Longrightarrow B$
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$\underbrace{abs}_{test}$
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sub and superscript
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$\beta_0$
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$\sum_{i=1}^{n} i$
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In an equation:
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\begin{equation}
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\sum_{j=1}{n} j^2 \beta
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\end{equation}
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Equation without number
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\begin{equation*}
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A \Rightarrow B
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\end{equation*}
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\section{Empirical Investigation}
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Here is some example code to create tables and graphs from the
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Python script. In order for this to work you would first need
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to run the script non\_linear\_models\_example\_report.py. Running that
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file (using the recommended file structure) creates some figures
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in the figures folder and some tables in .tex files in the report folder.
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\subsection{Question 3}
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For instance, here the file df\_table.tex is used print the actual numbers
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in the table.
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\begin{table}[h]
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\input{df_table}
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\caption{This tables has the estimates for ...}
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\label{tab::estimation_results}
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\end{table}
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\subsection{Question 4: Some graphs}
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\begin{figure}
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\includegraphics[width=0.6\paperwidth]{../figures/quadratic_model_y}
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\caption{This is a Figure coming straight from Python.}
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\label{fig::example_data}
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\end{figure}
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In Figure \ref{fig::example_data} we see the data.
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\begin{figure}
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\includegraphics[width=0.6\paperwidth]{../figures/quadratic_model_linear}
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\caption{This is a linear fit on a quadratic model.}
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\label{fig::example_quadratic_linear}
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\end{figure}
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In Figure \ref{fig::example_quadratic_linear} we see a linear fit.
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\begin{figure}
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\includegraphics[width=0.6\paperwidth]{../figures/quadratic_model_quadratic}
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\caption{This is quadratic fit on a quadratic model.}
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\label{fig::example_quadratic_quadratic}
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\end{figure}
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In Figure \ref{fig::example_quadratic_quadratic} we see that
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\subsection{Question 5}
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Equation example with matrices:
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\begin{equation}\label{eq::wald_test}
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H_0: \beta_1 = - \beta_2; \beta_3=0; \beta_2 + 2\beta_4 = 2
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\quad H_1: \neg H_0
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\end{equation}
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can be written in matrix form as:
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\begin{equation}\label{eq::matrix_form}
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\begin{bmatrix}
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1 & 1 & 0 & 0 \\
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0 & 0 & 1 & 0 \\
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0 & 1 & 0 & 2
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\end{bmatrix}
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\begin{bmatrix}
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\beta_1 \\
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\beta_2 \\
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\beta_3 \\
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\beta_4
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\end{bmatrix} =
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\begin{bmatrix}
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0 \\
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0 \\
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2
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\end{bmatrix}
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\end{equation}
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In equation \eqref{eq::wald_test} we see that... and in equation \eqref{eq::matrix_form} we see that
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\subsection{Question 6}
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$\beta$
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\begin{table}
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\input{summary}
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\caption{This tables has the estimates summary}
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\label{tab::estimation_results_summary}
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\end{table}
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Table \ref{tab::estimation_results_summary} has the full summary.
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\begin{table}
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\input{results_coef}
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\caption{This tables has the estimates summary}
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\label{tab::estimation_results_coef}
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\end{table}
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Table \ref{tab::estimation_results_coef} has the only the coefficient
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results.
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\end{document}
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