question 2.1 issue
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@ -138,7 +138,7 @@ In figure \ref{fig::plot_1_1} we have a 3D representation of the generated model
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We now estimate the parameters of $\beta_0$, $\beta_1$ and $\beta_2$ using the \textbf{Ordinary Least Squares} (OLS) method. With model:
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\[y_i=\beta_0 + \beta_1 x_1 + \beta_2 x2 + u_i\]
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\begin{table}[h]
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\begin{table}[ht]
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\centering
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\input{table_1_2}
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\caption{Linear Fit on Generated Data}
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@ -154,7 +154,7 @@ We can explain the bias of $\beta_0$ because this new model has a new error term
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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.
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\begin{table}[h]
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\begin{table}[ht]
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\centering
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\input{table_1_3}
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\caption{Linear Fit with 1 Variable}
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@ -165,7 +165,7 @@ Wich model would you choose? If I have sufficient calculation power, I would cho
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In figure \ref{fig::plot_1_4} we have a 3D representation of the generated model.
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\begin{figure}[h]
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\begin{figure}[ht]
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\includegraphics[width=0.6\paperwidth]{../figures/question_1_4}
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\caption{Generated points for Question 1.4.}
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\label{fig::plot_1_4}
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@ -173,7 +173,7 @@ In figure \ref{fig::plot_1_4} we have a 3D representation of the generated model
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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 $x_2^{new}$.
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\begin{table}[h]
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\begin{table}[ht]
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\centering
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\input{table_1_4}
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\caption{New Linear Fit on Generated Data}
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@ -185,7 +185,7 @@ The estimation results compared to the results in question 1.2 are similar, ther
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Similar as in question 1.3 we estimated the parameter with a single independent variable.
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\begin{table}[h]
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\begin{table}[ht]
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\centering
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\input{table_1_5}
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\caption{Linear Fit with 1 Variable}
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@ -225,7 +225,7 @@ Now we replace $x_1$ in the original model with
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\[x_1 \sim \mathcal{N}(3,\,1)\]
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If we now estimate the parameters we find:
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\begin{table}[h]
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\begin{table}[ht]
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\centering
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\input{table_1_6}
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\caption{Generate Data with Small Variance on x1}
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@ -234,7 +234,7 @@ If we now estimate the parameters we find:
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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 $x_2$.
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\begin{figure}[h]
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\begin{figure}[ht]
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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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@ -250,7 +250,7 @@ We can explain the difference in standard error of the estimates of $\beta_1$ us
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\[Var(\beta_1) = \sigma^2/Var(x_1)\]
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This means that $Var(\(beta_1) \sim 1/Var(x_1)$.
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This means that $Var(\beta_1) \sim 1/Var(x_1)$.
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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.
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@ -299,24 +299,21 @@ A \Rightarrow B
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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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\subsection{Question 2.1}
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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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\begin{table}[ht]
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\centering
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\input{summary_stats}
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\caption{Generate Data with Small Variance on x1}
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\label{tab::table_2_1}
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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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@ -325,24 +322,6 @@ in the table.
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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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@ -45,9 +45,9 @@ if not os.path.exists(report_dir):
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# first birthday
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bd_1 = 3112
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bd_1 = 303
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# second birthday
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bd_2 = 3112
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bd_2 = 309
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group_seed = bd_1 * bd_2
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@ -89,14 +89,23 @@ data = data_full.iloc[observations , :].copy()
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print_question('Question 2.1: Descriptive Statistics')
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# compute the summary statistics
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# data_summary = TODO
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data.drop(['fail', 'urban', 'unearn','househ', 'amtland', 'unearnx'],
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axis='columns',
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inplace=True)
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data = data[data['paidwork']==1]
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data['school'] = data['yprim']+data['ysec']
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data['wage'] = np.exp(data['lwage'])
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data_summary = data.describe()
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# print to screen
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# print(data_summary.T) [uncomment]
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print(data_summary.T)
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# export the summary statistics to a file
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# data_frame_to_latex_table_file(report_dir + 'summmary_stats.tex',
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# data_summary.T) [uncomment]
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data_frame_to_latex_table_file(report_dir + 'summary_stats.tex',
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data_summary.T)
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# -----------------------------------------------------------------------------
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# Question 2.2
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