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\begin{document}

\newcommand{\blind}{0}
\newcommand{\tit}{\bf Analyses }
% 1) Shedding Light on Causal Inference: ...
% 2)
% Marginal Structural Models for Sequential Excursion Effects
% \newcommand{\tit}{\bf Dynamic longitudinal marginal structural models for optogenetic designs}

% nonparametric
% sequentially randomized trials
% sequential optogenetic designs
% DTR
% Positivity
% Nonparametric Causal Inference for Sequential Optogenetic Designs: Dynamic Regimes with Positivity Violations
% Nonparametric Causal Inference in

% long
% Nonparametric Causal Inference for Sequential Optogenetic Designs: Excursion Effects for Dynamic Regimes

% short
% Nonparametric Causal Inference for Optogenetic Designs: Sequential Excursion Effects for Dynamic Regimes

%

% treatment sequence/sequence excursion effects

% micro longitudinal effects
% excursion effects
% longitudinal excursion effects
% multi-timepoint excursion effects
% HR-MSMs
% (causal) contrasts of mean counterfactual of deterministic regimes
% treatment effects

\if0\blind

{\title{\tit}
% \thanks{The authors declare no conflicts.
% }

% \author{Gabriel Loewinger\thanks{G.L. and A.W.L. contributed equally to this work.} \thanks{National Institute of Mental Health, NIH. Email: gloewinger@gmail.com},
% \and Alexander W. Levis$^*$\thanks{Postdoctoral Researcher, Carnegie Mellon University. Email: alevis@cmu.edu},
% \and
% Francisco Pereira\thanks{National Institute of Mental Health, NIH.}
% }

% \def\thefootnote{1}\footnotetext{G.L. and A.W.L. contributed equally to this work.}
% \nnfootnote{G.L. and A.W.L. contributed equally to this work.}


\date{}

\maketitle
}\fi

\if1\blind
\title{\bf \tit}
\maketitle
\fi

\section{Methods}
We modeled deal size to understand the factors that encourage and discourage climate investment. Specifically, we estimated the degree to which \textit{Environmental urgency}, \textit{Economic conditions}, and \textit{Institutional stability} variables (operationally defined) are associated with deal size. We fit a collection of working linear models to estimate the best linear \textit{projection} of the treatment (policy instruments) on the outcome (deal size). That is, we fit a model that places structure on the mean of the outcome (deal size), but not on the variance; we quantified uncertainty and conducted inference with a robust variance estimator. We adjust for a small set of observed potential confounders (with linear terms). We followed up with an Autoregressive models of these models to understand how the above factors change the mean outcome given levels during previous years.

\subsection{Notation}
We denote the outcome variable, $\texttt{deal}_i$, as the size of the deal $i \in \{1,2,...,n\}$. We denote $\texttt{country}_i$ as the country where the invested company of $\texttt{deal}_i$ was located and $\texttt{year}_i$ as the year that $\texttt{deal}_i$ took place.  For deal $i$, we denote the following variables for the corresponding country: $\texttt{CVI}_i$ is that country's climate vulnerability index, $\texttt{ease}_i$ is its Ease of Doing Business score, $\texttt{interest}_i$ is the interest rate, $\texttt{CPI}_i$ is the Consumer Price Index, $\texttt{GNI}_i$ is the Gross National Income, $\texttt{PSI}_i$ is the Political Stability Index, $\texttt{corruption}_i$ is the XXX corruption score,  $\texttt{ER}_i$ is the Exchange Rate. We included a separate intercept (fixed effect) in the model for each year from 2000-2020. (i.e., we included year as a factor variable in the model). To avoid cumbersome notation, we denote the vector of year-specific indicator variables (to represent the different levels of the factor variable \texttt{year}) as $\mathbf{{year}}_i$, and the corresponding vector of regression coefficients as $\tilde{\boldsymbol{\beta}}$. In shorthand, we denote the vector of variables of interest as $\mathbf{A}_i$, the set of variables we adjust for as $\mathbf{X}_i$, and the set of coefficients with a superscript (e.g.,  $\boldsymbol{\beta}^a$, $\boldsymbol{\beta}^x$).

\subsection{Models}
\paragraph{Model 1} We fit the following working linear model to estimate the association between the intensity of the policy and the amount of deal size:
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align*} \label{mod1}
   \mathbb{E} \left [ \texttt{deal}_i \mid \mathbf{A}_i, \mathbf{X}_i, \mathbf{M}_i \right ] =& { \color{gray} \beta_0} + {\color{blue} \beta_1 \texttt{CVI}_i + \beta_2 \texttt{ease}_i + \beta_3 \texttt{interest}_i + \beta_4 \texttt{CPI}_i} + \\ & {\color{blue}  \beta_5 \texttt{enterprise}_i + \beta_6 \texttt{PSI}_i + \beta_7 \texttt{corrupt}_i} +\\ & {\color{blue} \beta_8 \texttt{GNI}_i + \beta_{9} \texttt{interest}_i +
     \beta_{10} \texttt{unemploy}_i} +\\
    & { \color{gray} \beta_{11} \texttt{ER}_i +
\mathbf{{year}}_{i}^T \tilde{\boldsymbol{\beta}} }.
\end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%%
We express {\color{blue} the main variables of interest in blue}, and {\color{gray} the variables we are adjusting for in gray}. We write this same model using the shorthands above as \newline
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align*} \label{mod1}
   \mathbb{E} \left [ \texttt{deal}_i \mid \mathbf{A}_i, \mathbf{X}_i \right ] =& \beta_0 + {\color{blue} \mathbf{A}_i^T \boldsymbol{\beta}^a } + {\color{gray} \mathbf{X}_i^T \boldsymbol{\beta}^x }. \tag{1}
\end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%%
The inferential targets were the coefficients ${\color{blue} \boldsymbol{\beta}^a} \in \mathbb{R}^{10}$. \newline

\paragraph{Model 2} We next sought to understand how the association between the variables in the vector $\mathbf{A}_i$ are \textit{modified} by United Nations developed/developing country status. We sought to test whether investment patterns in developing country businesses may be driven by negative perceptions unrelated to the Enviornmental Urgency, Economic conditions, and Institutional stability variables. If investors are only avoiding deals due to, for example, institutional instability, then one would expect the magnitude of the deal size–institutional stability association to be comparable in developed and developing countries. Conversely, if investors are avoiding investment due to, for example, negative perceptions of those countries, one would anticipate a statistically significant difference in the magnitude of associations between developed/developing countries. Thus we fit a model similar to that in expression \ref{mod1}, except that we include pairwise interactions between $\texttt{dev}_i$ and all variables in $\mathbf{A}_i$. These interaction terms (denoted below as $\boldsymbol{\gamma}$) thus provide one way to formalize our question into a hypothesis test (albeit with the caveat that the variables $\mathbf{A}_i$ may only capture some financial reasons why investors may be hesitant).

Specifically, we denote $\texttt{dev}_i = 1$ if $\texttt{country}_i$ is ``developed'' and 0 otherwise. We write the pairwise multiplication of the terms in the interaction as $\texttt{dev}_i * \mathbf{A}_i$, and the associated regression coefficients as $\boldsymbol{\gamma}$. We include a main effect of $\texttt{dev}_i$, but not $\texttt{country}_i$ because we sought to understand variability across countries, not within-country. The targets of inference in this model were the interaction coefficients, $\boldsymbol{\gamma}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{align*} \label{mod1.2}
   \mathbb{E} \left [ \texttt{deal}_i \mid \mathbf{A}_i, \mathbf{X}_i, \texttt{dev}_i \right ] =& \beta_0 + {\color{red} \texttt{dev}_i \beta_1 +} {\color{blue}   \mathbf{A}_i^T \boldsymbol{\beta}^a } + {\color{red}   \left( \texttt{dev}_i *\mathbf{A}_i \right)^T \boldsymbol{\gamma}} + {\color{gray} \mathbf{X}_i^T \boldsymbol{\beta}^x }  \tag{2}
\end{align*}
%%%%%%%%%%%%%%%%%%%%%%%%%%

% \paragraph{Model 2}
% We next model the deal-size amount as a function of a set of policy instruments, where the instrument $l$ is denoted as $\texttt{policy}_l \in \{0,1\}$ and $\mathbf{A}_i = [\texttt{policy}_1~~\texttt{policy}_2~~ \ldots ~~\texttt{policy}_L]$. To avoid confusion, we denote the regression coefficients for Models 2 and 2.2 as $\boldsymbol{\alpha}$ and $\tilde{\boldsymbol{\alpha}}$.
% %%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{align*} \label{mod2}
%    \mathbb{E} \left [ \texttt{deal}_i \mid \mathbf{A}_i, \mathbf{X}_i, \mathbf{M}_i \right ] =& \alpha_0 + \mathbf{A}_i^T \boldsymbol{\alpha}^a +  \mathbf{M}_i^T \boldsymbol{\alpha}^m + \mathbf{X}_i^T \boldsymbol{\alpha}^x \tag{2}
% \end{align*}
% %%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%
% \paragraph{Model 2.2} Similar to above, we fit a related model that without potential mediators.
% %%%%%%%%%%%%%%%%%%%%%%%%%%
% %%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{align*} \label{mod2.2}
%    \mathbb{E} \left [ \texttt{deal}_i \mid \mathbf{A}_i, \mathbf{X}_i \right ] =& \tilde{\alpha}_0 + \mathbf{A}_i^T \tilde{\boldsymbol{\alpha}}^a +  \mathbf{X}_i^T \tilde{\boldsymbol{\alpha}}^x \tag{2.2}
% \end{align*}
% %%%%%%%%%%%%%%%%%%%%%%%%%%
% \paragraph{Model 5}
% We next model the policy intensity score, $\texttt{intensity}_i$ as a function of the potential confounders and mediators. To avoid confusion, we denote the regression coefficients for Models 5 and 5.2 as $\boldsymbol{\gamma}$. For Model 5, we include the vector of covariates, $\tilde{\mathbf{X}}_i$, which contains the following variables: $\texttt{CPI}_i$, $\texttt{interest}_i$, $\texttt{unemploy}_i$, $\texttt{GDP}_i$, $\texttt{HDI}_i$, $\texttt{sector}_{i,j}$, $\texttt{year}_{i,k}$.
% %%%%%%%%%%%%%%%%%%%%%%%%%%
% \begin{align*} \label{mod5}
%    \mathbb{E} \left [ \texttt{intensity}_i \mid \tilde{\mathbf{X}}_i \right ] =& \gamma_0 + \tilde{\mathbf{X}}_i^T \boldsymbol{\gamma}^x. \tag{5}
% \end{align*}
% %%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Regression Approach}
We fit \textit{projection}-based semi-parametric regressions with the above working models. We use a robust (sandwich) variance estimator (``CR2'')  \citep{CR2_sandwich} for inference with the $\texttt{sandwich}$ package in R \citep{sandwich_package}. We specified clustering at the country level to account for the longitudinal structure of the data because we anticipate outcomes $\texttt{deal}_i$ are correlated across deals within a country. However, this variance estimator is valid regardless of whether we mispecified the working correlation structure. Together this approach allows us to formalize our scientific questions within a null hypothesis testing framework with fewer assumptions about the distribution of $\texttt{deal}_i \mid \mathbf{A}_i, \mathbf{X}_i$.
% We perform multiple-comparison corrections when needed.

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