\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{Question 1.1} We investigate a linear model with noise \[y=\beta_0 + \beta_1 x1 + \beta_2 x2 + u\] where \[x1 \sim \mathcal{N}(3,\,36)\] \[x2 \sim \mathcal{N}(2,\,25)\] \[u \sim \mathcal{N}(0,\,9)\] 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{Question 1.2} We now estimate the parameters of $\beta_0$, $\beta_1$ and $\beta_2$ using the \textbf{Ordinary Least Squares} (OLS) method. With model: \[y_i=\beta_0 + \beta_1 x_1 + \beta_2 x2 + u_i\] \begin{table}[ht] \centering \input{table_1_2} \caption{Linear Fit on Generated Data} \label{tab::table_1_2} \end{table} In table 1 we can see that those estimates are are close to their true values within 2\%. Because our estimation model is the same as the model used to generate the data, we have a sufficient number of points and the assumptions of OLS are satisfied. In this situation we can expect good results of OLS estimation. \subsection{Question 1.3} If we compare the estimates with those of question 1.2. We see that the estimate of the intersect is not close to the true value with a difference of 4. We can explain the bias of $\beta_0$ because this new model has a new error term: $\beta_2x_{2i} + u_i$. This error term no longer has an expacted value of 0, but in fact $\beta_2E(x_{2i}) + E(u_i)= 4$ wich is very close to the bias we find. For $\beta_1$, there is little to no bias. This can we explained because $x_2$ and $u$ are stochasticaly independent from $x_1$. The standard error is bigger because $\beta_2E(x_{2i}) + E(u_i)$ has a bigger variance than just $u$. Wich model would you choose? If I have sufficient calculation power, I would choose model 1.2 as it is much more accurate. However for a very resource constraint situation model 1.3 might give acceptable estimates. \begin{table}[ht] \centering \input{table_1_3} \caption{Linear Fit with 1 Variable} \label{tab::table_1_3} \end{table} \subsection{Question 1.4} In figure \ref{fig::plot_1_4} we have a 3D representation of the generated model. \begin{figure}[ht] \includegraphics[width=0.6\paperwidth]{../figures/question_1_4} \caption{Generated points for Question 1.4.} \label{fig::plot_1_4} \end{figure} The estimation results compared to the results in question 1.2 are similar, there is very little bias. It appears that $x_2^{new}$ is sufficiently independent from $x_1$. We expected very little bias because $x2_{new}$ has a large independent part compared to $x_1$. The standard errors of the estimates of $\beta_1$ and $\beta_2$ are about 25\% higher wich can be explained partly bij the lower standard deviation in $x_2^{new}$. \begin{table}[ht] \centering \input{table_1_4} \caption{New Linear Fit on Generated Data} \label{tab::table_1_4} \end{table} \subsection{Question 1.5} Similar as in question 1.3 we estimated the parameter with a single independent variable. \begin{table}[ht] \centering \input{table_1_5} \caption{Linear Fit with 1 Variable} \label{tab::table_1_5} \end{table} The OLS estimators for the slope coefficients are biased. We see that $\beta_1$ is $-3$ instead of the true value of $-4$. We can explain this bias in the following way, lets start from the model. \[y^{new}=\beta_0 + \beta_1 x_1 + \beta_2 x_2^{new} + u_i\] We now have: \[x_2^{new} = 0.5 * x_1 + x_2^{'}\] Where: \[x_2^{'}\sim \mathcal{N}(5,\,16)\] Substituting in the model: \[\Longrightarrow y^{new}=\beta_0 + \beta_1 x_1 + \beta_2(0.5 * x_1 + x_2^{'}) + u_i\] Lets fill in the betas with the actual values: \[\Longrightarrow y^{new}= 3 + -4 x_1 + 2(0.5 * x_1 + x_2^{'}) + u_i\] \[\Leftrightarrow y^{new}= 3 - 4 x_1 + x_1 + 2x_2^{'}) + u_i\] \[\Leftrightarrow [y^{new}= 3 - 3 x_1 + 2x_2^{'}) + u_i\] Here we can see in table \ref{tab::table_1_5} easily that the OLS estimator will find -3 as the estimate for $\beta_1$. Similarly as in question 1.3 we can explain the bias on the intercept. \subsection{Question 1.6} Now we replace $x_1$ in the original model with \[x_1 \sim \mathcal{N}(3,\,1)\] If we now estimate the parameters we find: \begin{table}[ht] \centering \input{table_1_6} \caption{Generate Data with Small Variance on x1} \label{tab::table_1_6} \end{table} We find in table \ref{tab::table_1_6} that the parameters are essentially unbiased but have a bigger standard error for the intersect and $\beta_1$. The standard error of $\beta_1$ is 6 times bigger (from 0.016 to 0.10). We see no difference of the estimates $\beta_2$. Because nothing has changed in $x_2$. \begin{figure}[ht] \includegraphics[width=0.6\paperwidth]{../figures/question_1_6} \caption{Generated points for Question 1.6.} \label{fig::plot_1_6} \end{figure} We expected a similar estimation result as in 1.2 because there are no changes except of the standard deviation of $x_1$. This means that the OLS assumptions are equally valid and we expect unbiased estimates. We can explain the difference in standard error of the estimates of $\beta_1$ using the formula of $Var(\beta_1)$. \[Var(\beta_1) = \sigma^2(X^tX)_{11}^{-1}\] We can write this as \[Var(\beta_1) = \sigma^2/Var(x_1)\] This means that $Var(\beta_1) \sim 1/Var(x_1)$. Because $Var(x_1)$ changed from 36 in to 1, we expect the standard error to be $/sqrd(36) = 6$ times bigger. Which is exactly what we found. If the standard deviation from $x_1$ changes to 0, $\beta_1$ cannot we calculated. As we have seen with the no multicollinearity assumption. \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} \subsection{Question 2.1} For instance, here the file df\_table.tex is used print the actual numbers in the table. \begin{table}[ht] \centering \input{summary_stats} \caption{Generate Data with Small Variance on x1} \label{tab::table_2_1} \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} \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}