347 lines
10 KiB
TeX
347 lines
10 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{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}
|
|
|
|
We retain 2510 observations.
|
|
|
|
\begin{table}[ht]
|
|
\centering
|
|
\input{summary_stats}
|
|
\caption{Generate Data with Small Variance on x1}
|
|
\label{tab::summary_stats}
|
|
\end{table}
|
|
|
|
|
|
\subsection{Question 2.2}
|
|
|
|
\begin{figure}
|
|
\includegraphics[width=0.6\paperwidth]{../figures/question_2_2_wage}
|
|
\caption{Histogram wage}
|
|
\label{fig::question_2_2_wage}
|
|
\end{figure}
|
|
|
|
\begin{figure}
|
|
\includegraphics[width=0.6\paperwidth]{../figures/question_2_2_lwage}
|
|
\caption{Histogram lwage}
|
|
\label{fig::question_2_2_lwage}
|
|
\end{figure}
|
|
|
|
The lwage histogram in fig \ref{fig::question_2_2_lwage} is nicely centered so there is no need to remove any outliners. This is also close to a normal distribution. The wage historgam in fig \ref{fig::question_2_2_wage} is not symmetrical but is leaning to the left. Clealy not normal distributed.
|
|
|
|
\subsection{Question 2.3}
|
|
|
|
\begin{table}[ht]
|
|
\centering
|
|
\input{table_2_3}
|
|
\caption{Correlation matrix}
|
|
\label{tab::table_2_3}
|
|
\end{table}
|
|
|
|
We can see that there is a positive correlation between wage and school. It means that people who go longer to school will get a higher wage. There is a negative correlation between age and school. The younger generation is higher educated than older generation. Chinese citizens are better payed than malay, indian citizens have a negative correlation with wage.
|
|
|
|
\subsection{Question 2.4}
|
|
|
|
|
|
|
|
\end{document}
|