Institute for Quantitative Social Science, Harvard University

Konstantin Kashin is a Fellow at the Institute for Quantitative Social Science at Harvard University and will be joining Facebook's Core Data Science group in September 2015. Konstantin develops new statistical methods for diverse applications in the social sciences, with a focus on causal inference, text as data, and Bayesian forecasting. He holds a PhD in Political Science and an AM in Statistics from Harvard University.

Directed acyclic graphs (DAGs) are commonly used to represent causal relationships between variables across a wide variety of disciplines. For an excellent (and quite accessible) textbook on the topic, please see the book *Causal Inference* by Miguel Hernan and Jamie Robins.

In this post, I briefly explore how you can draw DAGs in LaTeX. In the subsequent post, I will show how to draw DAGs using R.

The first example of DAG is the common instrumental variable (IV) setup:

We seek to identify the effect of treatment *T* on outcome *Y*. However, this is confounded by an unmeasured variable *U*. The IV is denoted as *Z*. Technically, we do not need to put in the crossed out red edges between *U* and *Z* and *Z* and *Y* - absence of edges encodes independence relations. However, I included them to reinforce (and make explicit) the assumptions needed for identification of causal effects using IVs:

- (Quasi)-exogeneity of the instrument (no path from
*U*to*Z*) - Exclusion restriction (no direct path from
*Z*to*Y*)

This was made using the TikZ package in LaTeX. I used the `\usepackage{pgf,tikz}`

command at the beginning of my document.

The code to create the DAG above is:

```
\begin{tikzpicture}
% nodes %
\node[text centered] (z) {$Z$};
\node[right = 1.5 of z, text centered] (t) {$T$};
\node[right=1.5 of t, text centered] (y) {$Y$};
\node[draw, rectangle, dashed, above = 1 of t, text centered] (u) {$U$};
% edges %
\draw[->, line width= 1] (z) -- (t);
\draw [->, line width= 1] (t) -- (y);
\draw[->,red, line width= 1,dashed] (u) --node {X} (z);
\draw[->,line width= 1] (u) --(t);
\draw[->,line width= 1] (u) -- (y);
\draw[->, red, line width=1,dashed] (z) to [out=270,in=270, looseness=0.5] node{X} (y);
\end{tikzpicture}
```

Note that I first create the nodes (corresponding to variables in the DAG), and then draw the directed edges between the nodes.

Another example of a DAG is a simple structural equation model where we want each edge to be marked with the parameter signifying the causal effect. For example:

In LaTeX:

```
\begin{tikzpicture}
% nodes %
\node[text centered] (t) {$T$};
\node[right = 1.5 of t, text centered] (m) {$M$};
\node[right=1.5 of m, text centered] (y) {$Y$};
% edges %
\draw[->, line width= 1] (t) -- node[above,font=\footnotesize]{$\beta$} (m);
\draw [->, line width= 1] (m) -- node[above,font=\footnotesize]{$\gamma$} (y);
\draw[->, line width=1] (t) to [out=270,in=270, looseness=0.5] node[below, font=\footnotesize]{$\delta$} (y);
\end{tikzpicture}
```