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alex 2022-12-05 17:31:40 -07:00
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Defense_proposal/Sections/Oil_Econometrics.tex
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@ -24,29 +24,128 @@ Where \(Q_{O,b,t}\) is the net present oil produced in a basin state pair b, at
\begin{frame}{Oil and Gas Production: Model Sequence of Events}
\begin{center}
\begin{tikzpicture}
\draw [thick,decorate,decoration={brace,amplitude=6pt,raise=0pt}] (0.25,0.15) -- (3.25,0.15);
1.25
\node[align=center] at (2.4,0.55) {t};
\node[align=center] at (0,0.95) {\(Y_{t-1}\)};
\draw [thick,->] (0,.75) -- (0,0.15);
\node[align=center] at (2.25,.95) {\(Q_{energy}\)};
\draw [thick,->] (2.25,.75) -- (2.25,0.15);
\draw [thick,->] (2.25,-0.75) -- (2.25,-0.15);
\node[align=center] at (2.25,-0.95) {\(\theta_{oil}\)};
\draw [thick,->] (1.5,0) -- (14,0);
\node[align=center] at (2.5,0.25) {\(Y_{t-1}\)};
%%%%%%%%%
%%%left
\node[align=center] at (6,2.25) {\(Q_{energy}\)};
\draw [thick,-] (6,1.75) -- (6,2);
%%%right
\node[align=center] at (4,2.25) {\(\theta_{oil}\)};
\draw [thick,-] (4,1.75) -- (4,2);
%%Middle line
\draw [thick,->] (5,1.75) -- (5,0.15);
%%Down line
\draw [thick,-] (4,1.75) -- (6,1.75);
%%%%%%%%%%%%%%%%%%%%%%%
\tikzset{font=\footnotesize}
\node[align=center] at (5,-.25) {\(t\)};
\node[align=center] at (10,-.25) {\(t+1\)};
\node[align=left] at (5,3.25) {Shocks to oil supply \\Shocks to shared supply};
\tikzset{font=\normalsize}
\draw [thick,decorate,decoration={brace,amplitude=6pt,raise=0pt}] (4,2.5) -- (6,2.5);
%%%%%%%%%%%%%%%%%%%%%%
%%Mid brace
\draw [thick,decorate,decoration={brace,amplitude=6pt,raise=0pt,mirror}] (5,-0.5) -- (10,-0.5);
\tikzset{font=\footnotesize}
\node[align=left] at (7.75,-1.5) {Shocks to oil demand \\ Shocks to gas demand\\Shocks to general demand};
\tikzset{font=\normalsize}
\node[align=center] at (7.5,0.25) {\(Y_{t}\)};
\node[align=center] at (12.5,0.25) {\(Y_{t+1}\)};
%%%left
\node[align=center] at (6.5,1.25) {\(P_{oil}\)};
\draw [thick,-] (6.5,0.75) -- (6.5,1);
%%%mid
\node[align=center] at (7.5,1.25) {\(P_{gas}\)};
\draw [thick,-] (7.5,0.75) -- (7.5,1);
%%%right
\node[align=center] at (8.5,1.25) {\(D_{i}\)};
\draw [thick,-] (8.5,0.75) -- (8.5,1);
%%Down line
\draw [thick,-] (7.5,0.75) -- (7.5,0.45);
%%Middle line
\draw [thick,-] (6.5,0.75) -- (8.5,0.75);
%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%Mid brace
%%%left
\node[align=center] at (11,2.25) {\(Q_{energy}\)};
\draw [thick,-] (11,1.75) -- (11,2);
%%%right
\node[align=center] at (9,2.25) {\(\theta_{oil}\)};
\draw [thick,-] (9,1.75) -- (9,2);
%%Middle line
\draw [thick,->] (10,1.75) -- (10,0.15);
%%Down line
\draw [thick,-] (9,1.75) -- (11,1.75);
%%%%%%%%%%%%%%%%%%%%%%%
\node[align=center] at (3.15,0.55) {t+1};
\draw [thick,decorate,decoration={brace,amplitude=6pt,raise=0pt}] (2.5,0.15) -- (3.75,0.15);
\draw [thick,->] (4,0.85) -- (4,0.15);
\node[align=center] at (6.25,0.6) {Infectious period};
\draw [thick,->] (8.25,0.85) -- (8.25,0.15);
\node[align=center] at (9.15,0.5) {Immune\\stage};
\draw [thick,decorate,decoration={brace,amplitude=6pt,raise=0pt}] (4.25,0.15) -- (8,0.15);
\draw [thick,->] (0,0) -- (10,0);
\draw [thick,->] (4.6,-0.85) -- (4.6,-0.15);
\draw [thick,->] (7.65,-0.4) -- (7.65,-0.15);
\draw [thick,decorate,decoration={brace,amplitude=6pt,raise=0pt,mirror}] (4.75,-0.15) -- (7.5,-0.15);
\node[align=center] at (6.25,-0.85) {Symptomatic\\period};
\node[align=center] at (4.6,-1.3) {Onset of\\symptoms};
\end{tikzpicture}
\end{center}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Oil and Gas Production: Econometric Model }
\begin{block}{Supply Choices}
\begin{equation}
q_{t}=\rho_{11}Y_{t-1}+\cdots+\rho_{1\ell}Y_{t-\ell}+f_{w}(t)+\epsilon_{q_{t}}
\end{equation}
\begin{equation}
\theta_{t}=\rho_{21}Y_{t-1}+\cdots+\rho_{2\ell}Y_{t-\ell}+f_{\theta}(t)+\epsilon_{\theta_{t}}
\end{equation}
\end{block}
\begin{block}{Demand Shifters}
\begin{equation}
I_{t}=\rho_{31}Y_{t-1}+\cdots+\rho_{3\ell}Y_{t-\ell}+f_{I}(t)+\epsilon_{I}
\end{equation}
\begin{equation}
p_{g}=\beta_{43}I_{t}+\rho_{41}Y_{t-1}+\cdots+\rho_{4\ell}Y_{t-\ell}+f_{p_{g}}(t)+\epsilon_{p_{g}}
\end{equation}
\begin{equation}
p_{o}=\beta_{53}I_{t}+\beta_{54}P_{g}+\rho_{51}Y_{t-1}+\cdots+\rho_{5\ell}Y_{t-\ell}+f_{p_{o}}(t)+\epsilon_{p_{o}}
\end{equation}
\end{block}
\end{frame}
\begin{frame}{Oil and Gas Production: Econometric Model }
\[
\begin{pmatrix}
e_{t}^{\Delta Btu}\\
e_{t}^{\Delta \theta}\\
e_{t}^{\Delta I}\\
e_{t}^{\Delta p_{g}}\\
e_{t}^{\Delta p_{o}}
\end{pmatrix}
=
\begin{bmatrix}
a_{11} & 0 & 0 & 0 & 0\\
0 & a_{22} & 0 & 0 & 0 \\
0 & 0 & a_{33} & 0 & 0 \\
a_{41} & a_{42} & a_{43}& a_{44} & 0 \\
a_{51} & a_{52} & a_{53} & a_{54}& a_{55}
\end{bmatrix}
\begin{pmatrix}
\epsilon_{q}\\
\epsilon_{\theta}\\
\epsilon_{I}\\
\epsilon_{g}\\
\epsilon_{o}
\end{pmatrix}
\begin{matrix}
\textrm{Drilling supply shock} \\
\textrm{Composition supply shock} \\
\textrm{Aggregate demand shock} \\
\textrm{Gas specfic demand shock} \\
\textrm{Oil specfic demand shock}
\end{matrix}
\]
\end{frame}