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question 2.1 issue

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Hendrik Tillemans 2024-12-30 21:01:10 +01:00
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report/Assignment.tex
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@ -138,7 +138,7 @@ In figure \ref{fig::plot_1_1} we have a 3D representation of the generated model
We now estimate the parameters of $\beta_0$, $\beta_1$ and $\beta_2$ using the \textbf{Ordinary Least Squares} (OLS) method. With model: We now estimate the parameters of $\beta_0$, $\beta_1$ and $\beta_2$ using the \textbf{Ordinary Least Squares} (OLS) method. With model:
\[y_i=\beta_0 + \beta_1 x_1 + \beta_2 x2 + u_i\] \[y_i=\beta_0 + \beta_1 x_1 + \beta_2 x2 + u_i\]
\begin{table}[h] \begin{table}[ht]
\centering \centering
\input{table_1_2} \input{table_1_2}
\caption{Linear Fit on Generated Data} \caption{Linear Fit on Generated Data}
@ -154,7 +154,7 @@ We can explain the bias of $\beta_0$ because this new model has a new error term
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. 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}[h] \begin{table}[ht]
\centering \centering
\input{table_1_3} \input{table_1_3}
\caption{Linear Fit with 1 Variable} \caption{Linear Fit with 1 Variable}
@ -165,7 +165,7 @@ Wich model would you choose? If I have sufficient calculation power, I would cho
In figure \ref{fig::plot_1_4} we have a 3D representation of the generated model. In figure \ref{fig::plot_1_4} we have a 3D representation of the generated model.
\begin{figure}[h] \begin{figure}[ht]
\includegraphics[width=0.6\paperwidth]{../figures/question_1_4} \includegraphics[width=0.6\paperwidth]{../figures/question_1_4}
\caption{Generated points for Question 1.4.} \caption{Generated points for Question 1.4.}
\label{fig::plot_1_4} \label{fig::plot_1_4}
@ -173,7 +173,7 @@ In figure \ref{fig::plot_1_4} we have a 3D representation of the generated model
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 $x_2^{new}$. 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}[h] \begin{table}[ht]
\centering \centering
\input{table_1_4} \input{table_1_4}
\caption{New Linear Fit on Generated Data} \caption{New Linear Fit on Generated Data}
@ -185,7 +185,7 @@ The estimation results compared to the results in question 1.2 are similar, ther
Similar as in question 1.3 we estimated the parameter with a single independent variable. Similar as in question 1.3 we estimated the parameter with a single independent variable.
\begin{table}[h] \begin{table}[ht]
\centering \centering
\input{table_1_5} \input{table_1_5}
\caption{Linear Fit with 1 Variable} \caption{Linear Fit with 1 Variable}
@ -225,7 +225,7 @@ Now we replace $x_1$ in the original model with
\[x_1 \sim \mathcal{N}(3,\,1)\] \[x_1 \sim \mathcal{N}(3,\,1)\]
If we now estimate the parameters we find: If we now estimate the parameters we find:
\begin{table}[h] \begin{table}[ht]
\centering \centering
\input{table_1_6} \input{table_1_6}
\caption{Generate Data with Small Variance on x1} \caption{Generate Data with Small Variance on x1}
@ -234,7 +234,7 @@ If we now estimate the parameters we find:
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 $x_2$. 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}[h] \begin{figure}[ht]
\includegraphics[width=0.6\paperwidth]{../figures/question_1_6} \includegraphics[width=0.6\paperwidth]{../figures/question_1_6}
\caption{Generated points for Question 1.6.} \caption{Generated points for Question 1.6.}
\label{fig::plot_1_6} \label{fig::plot_1_6}
@ -250,7 +250,7 @@ We can explain the difference in standard error of the estimates of $\beta_1$ us
\[Var(\beta_1) = \sigma^2/Var(x_1)\] \[Var(\beta_1) = \sigma^2/Var(x_1)\]
This means that $Var(\(beta_1) \sim 1/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. 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.
@ -299,24 +299,21 @@ A \Rightarrow B
\section{Empirical Investigation} \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} \subsection{Question 2.1}
For instance, here the file df\_table.tex is used print the actual numbers For instance, here the file df\_table.tex is used print the actual numbers
in the table. in the table.
\begin{table}[h] \begin{table}[ht]
\input{df_table} \centering
\caption{This tables has the estimates for ...} \input{summary_stats}
\label{tab::estimation_results} \caption{Generate Data with Small Variance on x1}
\label{tab::table_2_1}
\end{table} \end{table}
\subsection{Question 4: Some graphs} \subsection{Question 4: Some graphs}
\begin{figure} \begin{figure}
@ -325,24 +322,6 @@ in the table.
\label{fig::example_data} \label{fig::example_data}
\end{figure} \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} \subsection{Question 5}

21
scripts/empirical.py
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@ -45,9 +45,9 @@ if not os.path.exists(report_dir):
# first birthday # first birthday
bd_1 = 3112 bd_1 = 303
# second birthday # second birthday
bd_2 = 3112 bd_2 = 309
group_seed = bd_1 * bd_2 group_seed = bd_1 * bd_2
@ -89,14 +89,23 @@ data = data_full.iloc[observations , :].copy()
print_question('Question 2.1: Descriptive Statistics') print_question('Question 2.1: Descriptive Statistics')
# compute the summary statistics # compute the summary statistics
# data_summary = TODO
data.drop(['fail', 'urban', 'unearn','househ', 'amtland', 'unearnx'],
axis='columns',
inplace=True)
data = data[data['paidwork']==1]
data['school'] = data['yprim']+data['ysec']
data['wage'] = np.exp(data['lwage'])
data_summary = data.describe()
# print to screen # print to screen
# print(data_summary.T) [uncomment] print(data_summary.T)
# export the summary statistics to a file # export the summary statistics to a file
# data_frame_to_latex_table_file(report_dir + 'summmary_stats.tex', data_frame_to_latex_table_file(report_dir + 'summary_stats.tex',
# data_summary.T) [uncomment] data_summary.T)
# ----------------------------------------------------------------------------- # -----------------------------------------------------------------------------
# Question 2.2 # Question 2.2