



\subsection{Selection as a Function of Treatment Alone}
Consider the most basic scenario of IV selection bias in Figure \ref{fig:IVDAGBaseline}. 
As stated above, $ Z $ in this model is a valid instrumental variable for the causal effect of $ T $ on $ Y $, $ \beta $ , if the analysis does not condition on $ S $. 
Conditioning on $ S $, however, invalidates $ Z $ as an instrumental variable, because $ S $ is a descendant of $ T $, and $ T $ is a collider on the path $ Z\rightarrow T\leftarrow U\rightarrow Y $. 
Conditioning on $ S $ opens this path, which induces an association between $ Z $ and $ Y $ via $ U $ and hence violates the exclusion condition.
 
Proposition \ref{prop:IVAdjBias} gives the selection bias in the standard IV estimator when the analysis adjusts for S. 

\begin{proposition}\label{prop:IVAdjBias}
	In a linear and homogeneous model with normal errors represented by Figure \ref{fig:IVDAGBaseline} and covariate adjustment on $ S $, the standard instrumental variables estimator converges in probability to 
	\[ 	
	\beta_{IV|Adj}=\beta -\delta_1 \delta_2  \frac{\gamma^2}{1-\gamma ^2}.
	\]
\end{proposition}

The proof follows from regression algebra and Wright's rule (\citealt{Wright1934}). 
The magnitude of selection bias due to covariate adjustment in the IV estimator depends on two components. 
First, selection bias increases with the strength of unobserved confounding between $ T $ and $ Y $ via $ U $, $ \delta_1 \delta_2 $ (which corresponds to the path $ Z\rightarrow T\leftarrow U\rightarrow Y $ that is opened by conditioning on $ S $, less the first stage $ Z\rightarrow T $). 
Second, selection bias increases with the effect of the treatment $ T $ on the selection variable, $ S $ , $ \gamma $ . 
When $ \gamma =0 $, $ S $ contains no information about the collider $ T $, conditioning on $ S $ does not open the path $ Z\rightarrow T\leftarrow U\rightarrow Y $, and selection bias is zero. 
By contrast, as $ |\gamma |\rightarrow 1 $, the magnitude of the bias increases without bound because adjusting for $ S $ increasingly amounts to adjusting for the collider $ T $ itself, while at the same time reducing the first stage. 
(If the analysis directly adjusted for $ T $, then the first stage would go to zero and the IV estimator would not be defined.)

Proposition \ref{prop:IVTruncBias} derives the IV selection bias due to interval truncation on S.
\begin{proposition}\label{prop:IVTruncBias}
	In a linear and homogeneous model with normal errors represented by Figure \ref{fig:IVDAGBaseline} and truncation on $ S $, $ R=\mathbf{1}(S\geq s_0) $, the standard instrumental variables estimator converges in probability to 
	\[ 
	\beta_{IV|Tr}=\beta -\delta_1 \delta_2  \frac{\psi \gamma ^2}{1-\psi \gamma^2}, \quad \text{where }  \psi =\frac{\phi (s_0)}{1-\Phi(s_0)} \left( \frac{\phi (s_0 )}{1-\Phi(s_0 )} - s_0 \right),
	\]
	and $ \phi (\cdot) $ and $ \Phi(\cdot ) $ are the standard normal pdf and cdf, respectively. 
\end{proposition}

Proposition \ref{prop:IVTruncBias} (proved in Appendix \ref{sec:TruncBiasProof}) illustrates that IV selection bias due to truncation (Proposition \ref{prop:IVTruncBias}) differs from IV selection bias due to adjustment (Proposition \ref{prop:IVAdjBias}) only in that truncation deflates the contribution of the effect of $ T $ on $ S $, $ \gamma $, by the factor $ \psi \in (0,1) $.
Since $ \psi\ $  is the derivative of the standard normal hazard function, it monotonically increases with the \emph{severity of truncation}, $Pr(R=0)=\Phi(s_0)$, as shown in Figure \ref{fig:PsiVsSelection}. Hence, interval truncation leads to less IV selection bias than covariate adjustment in Figure \ref{fig:IVDAGBaseline},

\begin{figure}[t!]
	\centering
	\begin{subfigure}{0.48\linewidth}
	    \centering
	    \textbf{\qquad Truncation Severity versus $ \psi $}
		\includegraphics[width=\linewidth]{figures/PsiVsProbRSQ.png}
		\caption{}
		\label{fig:PsiVsSelection}
	\end{subfigure}
	\begin{subfigure}{0.50\linewidth}
	    \centering
	    \textbf{\qquad Least Biased Estimator}\\
		\includegraphics[width=\linewidth]{figures/IVvOLSBiasPreferenceProbRSQ.png}
		\caption{}
		\label{fig:LeastBiasEstPreference}
	\end{subfigure}
	\caption{(\subref{fig:PsiVsSelection}) $\psi$ monotonically increases with truncation severity. (\subref{fig:LeastBiasEstPreference}) Whether OLS or IV is less biased under selection depends on truncation severity and the effect of $ T $ on $ S $, $ |\gamma| $.}
	\label{fig:PsiPlots}
\end{figure}


\begin{corollary}\label{cor:IVTruncIVAdj}
	In a linear and homogeneous model with normal errors represented by Figure \ref{fig:IVDAGBaseline}, the magnitude of IV-adjustment bias is weakly larger than that of IV-truncation bias: 
	$
	\left| \beta_{IV|Adj} - \beta \right| \geq \left| \beta_{IV|Tr} - \beta \right|.
	$
\end{corollary}
Corollary \ref{cor:IVTruncIVAdj} makes intuitive sense. 
Adjustment involves first exactly stratifying and then averaging across strata defined by $ S=s $. 
Exact stratification on $ S $ uses all information about $ T $ that is contained in $ S $, hence opening the biasing path as much as conditioning on $ S $ possibly can.  
By contrast, interval truncation amounts to imprecise stratification on $ S $ (retaining observations across a range of values on $ S $, but not exactly stratifying on any particular value), hence ``less opening" the biasing path.

Of some methodological interest, we further note, in Figure \ref{fig:IVDAGBaseline}, that IV selection bias by truncation converges on IV selection bias by covariate adjustment as the severity of truncation increases to shrink the remaining sample to a single point. 
Proposition \ref{prop:AdjIsPtTrunc} states that this observation is true for all models, not only for Figure \ref{fig:IVDAGBaseline}.
\begin{proposition}\label{prop:AdjIsPtTrunc}
	In a linear and homogeneous model with normal errors, selection bias in the standard instrumental variables estimator due to covariate adjustment is the limiting case of selection bias due to point truncation, 
	\[ 
	\lim_{s_0\rightarrow \infty}\beta_{IV|Tr} = \beta_{IV|Adj}.   
	\]
\end{proposition}

This proposition makes intuitive sense. 
Covariate adjustment involves exact stratification on $ S=s $, which defines point truncation. 
Since the probability limits of all $s$-stratum specific estimators are identical in linear Gaussian models, selection bias by adjustment equals selection bias by point truncation.  
The proof in Appendix \ref{sec:TruncAdjProof} formalizes this intuition. 

Proposition \ref{prop:IVTruncBias} helps inform empirical choices in practice. When selection is unavoidable (e.g. because the data were truncated during data collection), should analysts choose IV or OLS? 
Figure \ref{fig:LeastBiasEstPreference} shows that the IV estimator is preferred to OLS, with respect to bias, for most combinations of $ \gamma  $ and truncation severity.
Since OLS bias (with or without truncation) only depends on unobserved confounding, i.e. $ \beta _{OLS|Tr} - \beta=\delta_1 \delta_2 $, the difference in magnitude between the OLS and IV biases with truncation is given by 
\[ 
\left|\beta_{OLS|Tr}-\beta\right| - \left|\beta_{IV|Tr} - \beta \right|=\left|\delta_1 \delta_2 \right|  \frac{1-2\psi \gamma^2}{1-\psi \gamma^2}.
\]
Hence, the IV estimator is preferred when $ \psi \gamma^2 \leq \frac{1}{2} $.
Specifically, when fewer than 29.1\% of observations are truncated (corresponding to $ \psi \leq 0.5 $), IV is preferred regardless of the effect of $ T $ on $ S $, $ \gamma $. 
Conversely, when $ |\gamma |<\sqrt{0.5}\approx 0.707 $, no amount of truncation makes OLS preferable over IV. 
Recalling that $ \gamma $ cannot exceed 1 in magnitude, the selection variable $ S $ would have to be an extraordinarily strong proxy for $ T $ to make IV more biased than OLS at any level of truncation.

Perhaps most useful for practice, we note that selection bias (by truncation or adjustment) in Figure \ref{fig:IVDAGBaseline} is proportional to the negative of OLS confounding bias. 
Therefore, the OLS and IV estimators under selection bound the true causal effect.

\begin{corollary}
	In a linear and homogeneous model with normal errors represented by Figure \ref{fig:IVDAGBaseline}, the OLS estimator and the instrumental variables estimator with selection bound the causal effect of $ T $ on $ Y $, $ \beta $ ,
	\begin{align*}
        \beta_{IV|Tr}\leq & \beta \leq \beta_{OLS}, \quad \text{when } \quad \delta_1 \delta_2>0, \\
	    \beta_{IV|Tr}\geq & \beta \geq \beta_{OLS}, \quad \text{when } \quad \delta_1 \delta_2<0. 	    
	\end{align*} 
\end{corollary}

The fact that the IV selection bias has the opposite sign of the OLS selection bias in Figure \ref{fig:IVDAGBaseline} is owed to linearity and homogeneity: in linear and homogeneous models, conditioning on a collider or its descendant reverses the sign of the product of the path parameters for the associated path. 
For example if all path parameters along the biasing path $ Z\rightarrow T\leftarrow U\rightarrow Y $ are positive, then conditioning on $ S\in desc(T) $ will induce a negative association along this path. 
Since the IV bias hinges on conditioning on $ S $, the selection bias would be negative. 
By contrast, OLS bias in Figure \ref{fig:IVDAGBaseline} does not hinge on conditioning on $ S $ and instead results from confounding along $ T\leftarrow U\rightarrow Y $. 
Therefore, OLS bias would be positive. 

\subsection{Selection as a Function of a Mediator}\label{sec:medselection}

Next, consider models in which the selection variable, $ S $, is a mediator of the effect of treatment on the outcome, as in the causal graphs in Figures \ref{fig:DAGTandY} and \ref{fig:DAGTandYandV}. 
These situations are worth investigating for two reasons: first, often empiricists are interested in the direct causal effect of $ T $ on $ Y $, which necessitates conditioning on $ S $; 
second, they result in qualitatively different bias representations.

\begin{figure}[b!]
	\centering
	\begin{subfigure}{0.48\linewidth}
		\input{figures/IVDAG_T_and_Y.tex}
		\caption{}
		\label{fig:DAGTandY}
	\end{subfigure}
	\begin{subfigure}{0.48\linewidth}
		\input{figures/IVDAG_T_and_Y_and_V.tex}
		\caption{}
		\label{fig:DAGTandYandV}
	\end{subfigure}
	\caption{IV scenarios where the selection variable is both a descendant of treatment and a mediator.}
	\label{fig:IVDAGMediated}
\end{figure}

Suppose that the analyst is interested in the direct causal effect of $ T $ on $ Y $, $ \beta $, in the model of Figure \ref{fig:DAGTandY}. 
The bias in the IV and OLS estimators under interval truncation and adjustment for $ S $ is given in Proposition \ref{prop:IVMedBias}. 

\begin{proposition}\label{prop:IVMedBias}
	In a linear and homogeneous model with normal errors represented by Figure \ref{fig:DAGTandY}. The standard instrumental variables estimator with selection on $ S $, converges in probability to 
	\[ 
	\beta_{IV|S}= \beta -\delta_1 \delta_2  \frac{\psi \gamma ^2}{1-\psi \gamma ^2} + \gamma\tau \frac{1-\psi}{1-\psi \gamma ^2},
	\]
	and the OLS estimator with selection on $ S $ converges in probability to 
	\[ 
	\beta_{OLS|S}= \beta +\delta_1 \delta_2+\gamma \tau \frac{1-\psi}{1-\psi \gamma ^2},
	\]
	where 
	\[ 
	\psi =\begin{cases}
				\frac{\phi (s_0)}{1-\Phi(s_0)} \left( \frac{\phi (s_0 )}{1-\Phi(s_0 )} - s_0 \right) & \text{with truncation on }S, R=\mathbf{1}(S\geq s_0) \\
				1 & \text{with adjustment on }S 
		\end{cases}.
	\]
\end{proposition}
All bias expressions in Proposition \ref{prop:IVMedBias} have a straightforward graphical interpretation. 
With \emph{adjustment} on $ S $, the indirect causal path $ T\rightarrow S\rightarrow Y $ is completely blocked, because $ S $ is a non-collider on this path. 
Hence, the bias in the IV and OLS estimators with adjustment on $ S $ equals the IV and OLS adjustment biases in Figure \ref{fig:IVDAGBaseline}, where $ S $ was not a mediator.
With adjustment on $ S $, IV is biased by selection, whereas OLS is biased by confounding; 
IV selection bias will generally be smaller in magnitude than OLS confounding bias (unless the effect of $ T $ on $ S $ is very large); and IV and OLS with adjustment bound the true direct causal effect. 

With \emph{truncation} on $ S $, however, the indirect path $ T\rightarrow S\rightarrow Y $ is not completely blocked and hence contributes a new term to both IV and OLS bias. For both IV and OLS, this term equals the strength of the partially blocked indirect path, $ \gamma \tau $ , deflated by the multiplier $ 0\leq(1-\psi)/(1-\psi \gamma^2 )\leq 1 $. 
The size of the multiplier depends both on the truncation severity, $ \psi $ , and on the effect of $ T $ on $ S $, $ \gamma $ , but in opposite directions.
As $ \gamma $ is fixed and truncation increases, $ \psi \rightarrow 1 $, the analysis conditions ever more precisely on an ever smaller range of values of $ S $; hence the indirect path is increasingly blocked, and both the multiplier and the bias term tend to 0. 
By contrast, when $ \psi $ is fixed and the effect of $ T $ on $ S $ increases, $ |\gamma| \rightarrow 1 $, the information about $ T $ contained in $ S $ increases, the multiplier tends to 1, and the path is increasingly opened.
 
By Proposition \ref{prop:AdjIsPtTrunc}, it remains true in Figure \ref{fig:DAGTandY} that IV selection bias due to adjustment is the limiting case of IV selection bias due to point truncation. 
However, it is no longer necessarily true that IV with adjustment is more biased than IV with truncation. 
The bias ordering now depends on the signs and relative sizes of the two additive bias term (representing the biasing paths $ T\leftarrow U\rightarrow Y $ and $ T\rightarrow S\rightarrow Y $), and on how well the indirect path $ T\rightarrow S\rightarrow Y $ is closed by truncation. 
Hence, when selection is made on a mediator of the treatment effect, selection bias by adjustment could be larger or smaller in magnitude than selection bias by truncation.
 Bounding the true causal effect also becomes more difficult. 
With truncation on $ S $, IV and OLS with selection do not necessarily bound the true direct causal effect. 

The analysis is further complicated when the effect of $ S $ on $ Y $ is confounded by some unobserved variable, $ W $, as in Figure \ref{fig:DAGTandYandV}. 
This situation is arguably more realistic than the model in Figure \ref{fig:DAGTandY}, because mediators in observational studies are expected to be confounded. 
Here, conditioning on $ S $ (by adjustment or truncation) in IV analysis opens a new path, $ Z\rightarrow T\rightarrow S\leftarrow W\rightarrow Y $, which violates the exclusion assumption; and in OLS it opens $ T\rightarrow S\leftarrow W\rightarrow Y $, which biases OLS regression. 
The resulting bias expressions are the same as those in Proposition \ref{prop:IVMedBias} with an additional bias term, $ -\gamma \delta_3 \delta_4  \frac{\psi}{1-\psi \gamma^2} $. 
Once more, IV selection bias due to adjustment is the limiting case of IV selection bias due to point truncation. 
However, no pair of estimators (among $ \beta_{IV|Tr}, \beta_{IV|Adj},\beta_{OLS|Tr},\beta_{OLS|Adj} $) can be relied on to bound the true direct causal effect in the model of Figure \ref{fig:DAGTandYandV}.

\subsection{Selection on Treatment and the Unobserved Confounder}\label{sec:medandconfselection}

Finally, we consider situations where the selection variable, $ S $, is also a descendant of the unobserved $ U $ that confounds the effect of treatment on the outcome (Figure \ref{fig:DAGTandU}).

\begin{figure}[t!]
	\centering
	\input{figures/IVDAG_T_and_U.tex}
	\caption{IV scenario where the selection variable is both a descendant of the treatment and the unobserved confounder.}
	\label{fig:DAGTandU}
\end{figure}


\begin{proposition}\label{prop:IVUnobsBias}
	In a linear and homogeneous model with normal errors represented by Figure \ref{fig:DAGTandU}. The standard instrumental variables estimator with selection on $ S $ converges in probability to
	\[ 	
	\beta_{IV|S} = \beta -\delta_1 \delta_2  \frac{\psi \gamma ^2}{1-\psi \gamma (\gamma +\delta_1 \delta_3 )} - \gamma\delta_3\delta_2\frac{\psi}{1-\psi \gamma (\gamma +\delta_1 \delta_3 )}, 
	\]
	and the OLS estimator with selection on S converges in probability to 
	\[ 
	\beta_{OLS|S} = \beta +\delta_1\delta_2  \frac{1-\psi  (\gamma^2+\gamma \delta_1 \delta_3+\delta_3^2)}{1-\psi\gamma(\gamma+\delta_1 \delta_3 )^2} - \gamma \delta_3 \delta_2\frac{\psi}{1-\psi \gamma (\gamma +\delta_1 \delta_3 )^2},
	\]
	where 
	\[ 
	\psi =\begin{cases}
	\frac{\phi (s_0)}{1-\Phi(s_0)} \left( \frac{\phi (s_0 )}{1-\Phi(s_0 )} - s_0 \right) & \text{with truncation on }S,R=\mathbf{1}(S\geq s_0) \\
	1 & \text{with adjustment on }S 
	\end{cases}.
	\]
\end{proposition}

Three points stand out about selection bias in Figure \ref{fig:DAGTandU}. 
First, when $ S $ is a descendant of both $ T $ and $ U $, conditioning on $ S $ opens a new path, $ T\rightarrow S\leftarrow U\rightarrow Y $, which biases IV and OLS with adjustment or truncation on $ S $.
 
 Second, in contrast to models considered previously, the bias term associated with each biasing path ($ T\leftarrow U\rightarrow Y $ and $ T\rightarrow S\leftarrow U\rightarrow Y $) is now a function of the path parameters of both paths.  In other words, the path-specific biases interact. Pearl's graphical causal models provide  intuition for this interaction.
Consider, for example, the second bias term. First, conditioning on $ S $ opens the path $ T\rightarrow S\leftarrow U\rightarrow Y $. Hence, the bias term depends on $ \gamma \delta_3 \delta_2 $. Second, conditioning on $ S $ also absorbs variance from $ U $ (a non-collider on $ T\rightarrow S\leftarrow U\rightarrow Y $), because $S$ is a descendant of $U$ along the path $U\rightarrow T\rightarrow S$.  Hence, the bias term also depends on $ \delta_1 $. 

Third, the direction of the interaction, and hence the overall bias, depends on the specific parameter values. This makes the bias order of these estimators fairly unpredictable and prevents generic recommendations for or against any one estimator. This ambiguity provides additional motivation for using exact bias formulas for sensitivity analysis. 
    



\subsection{Proof of Truncation Bias Expressions}\label{sec:TruncBiasProof}
We derive the bias under truncation by leveraging a result from  \citet{Tallis1965}.
\begin{lemma} \label{lem:Tallis}
	Let $V\in \mathbb{R}^k$ follow a multivariate normal distribution, $ V \sim N \left( 0, \Sigma \right) $, and define the truncated random vector $ \widetilde{V} = \{ v\in V: c'v \geq p \} $ with $ p\in \mathbb{R} $,$ c\in\mathbb{R}^k $, and $ |c|=1 $.
	Then the expectation and variance of the truncated random vector are given by
	\begin{align*}
	E\left[ \widetilde{V} \right] & = \Sigma c \kappa^{-1} \lambda \left( \frac{p}{\kappa}\right)\\
	Var\left( \widetilde{V} \right) & = \Sigma - \Sigma cc' \Sigma \kappa^{-2}\psi
	\end{align*}
	where $ \kappa = \left(c'\Sigma c \right)^{-1/2} $, $ \lambda(x) = \frac{\phi(x)}{1-\Phi(x)}$ is the hazard function of the standard normal distribution, and 
	\[
	\psi = \lambda \left( \frac{p}{\kappa}\right)\left( \lambda \left( \frac{p}{\kappa}\right) - \frac{p}{\kappa} \right).
	\]
\end{lemma}
Using properties of the standard normal hazard function it can be shown that $\psi$ is in fact the derivative of the hazard function.

\begin{proof}[Proof of Proposition \ref{prop:IVTruncBias}]
	Consider the model described by Figure \ref{fig:IVDAGBaseline}. 
	Since the idiosyncratic shocks are all normally distributed, all variables in the model are normally distributed.
	Specifically for vectors $ V = \begin{bmatrix} Z & U & T & S & Y \end{bmatrix}' $ and $ \varepsilon = \begin{bmatrix} \varepsilon_Z & \varepsilon_U & \varepsilon_T & \varepsilon_S & \varepsilon_Y \end{bmatrix}' $,
	 the standardized\footnote{Standardization implies non-unit variance for some of the shocks. For example when $ Var(T)=1 $, then $ \varepsilon_{T} $ is $ Var(\varepsilon_T) = 1-\pi^2-\delta_1^2 $.} model has the reduced form $ V = \Gamma \varepsilon $, where $ \varepsilon \sim N(0,\Sigma_{\varepsilon}) $ and
	\[ 
	\Gamma = \begin{bmatrix}
	1 & 0 & 0 & 0 & 0 \\
	0 & 1 & 0 & 0 & 0 \\
	\pi & \delta_{1} & 1 & 0 & 0 \\
	\gamma \pi & \gamma \delta_{1} & \gamma & 1 & 0 \\
	\beta \pi & \beta \delta_{1} + \delta_{2}& \beta & 0 & 1 \\
	\end{bmatrix} \qquad 
	\Sigma_{\varepsilon}= \begin{bmatrix}
	1 & 0 & 0 & 0 & 0 \\
	0 & 1 & 0 & 0 & 0 \\
	0 & 0 & 1-\pi^2-\delta_1^2 &  0 & 0 \\
	0 & 0 & 0 & 1-\gamma^2 &  0 \\
	0 & 0 & 0 & 0 & 1-\beta^2-\delta_2^2-2\beta\delta_1\delta_2 \\
	\end{bmatrix}.
	\]
	
	Since this implies that $ V \sim N\left(0, \Gamma \Sigma_{\varepsilon}\Gamma' \right) $, our truncation scenario, $ R = \mathbf{1}(S\geq s_0) $, allows for direct application of Lemma \ref{lem:Tallis} to derive the covariance matrix of the truncated distribution, $ \widetilde{V} = V|S\geq s_0 $.
	For Lemma \ref{lem:Tallis}, $ c= \begin{bmatrix} 0 & 0 & 0 & 1 & 0 \end{bmatrix}' $, $ p=s_0 $, and $ \Sigma = \Gamma \Sigma_{\varepsilon}\Gamma' $.
	This implies $ \kappa = 1 $ and thus
	\[ 
	Var\left( \widetilde{V} \right) = \Gamma \Sigma_{\varepsilon}\Gamma' - \Gamma \Sigma_{\varepsilon}\Gamma' cc' \Gamma \Sigma_{\varepsilon}\Gamma'\psi
	\quad \text{where} \quad \psi = \lambda( s_0 )\left( \lambda( s_0 ) - s_0 \right).
	\]
	
	Finally the IV estimand with truncation is given by the ratio of the truncated covariance between instrument and outcome and the truncated covariance between instrument and treatment. After some enjoyable algebra, we evaluate $ Var(\widetilde{V}) $, extract the relevant covariances, and obtain
	\begin{align*}
	\beta_{IV|Tr} & = \frac{Cov(Z,Y|S\geq s_0)}{Cov(Z,T|S\geq s_0)} = \frac{\beta\pi - \psi \gamma\pi\left( \beta\gamma + \gamma\delta_1\delta_2 \right)}{\pi - \psi\gamma^2\pi} = \beta - \delta_1\delta_2\frac{\psi\gamma^2}{1-\psi\gamma^2}.
	\end{align*}
\end{proof}

The proofs of Propositions \ref{prop:IVMedBias} and \ref{prop:IVUnobsBias} proceed analogously, using the appropriate reduced form matrix, $ \Gamma $, for each scenario.

\subsection{Proof of Adjustment as Point Truncation (Proposition \ref{prop:AdjIsPtTrunc})}\label{sec:TruncAdjProof}
\begin{proof}
    Define the stratum specific IV estimator when $S=s$ as 
    \[
    \beta_{IV|S}\left(s\right)=\frac{Cov\left(Z,Y|S=s\right)}{Cov\left(Z,T|S=s\right)}
    \]
    Notice that $\beta_{IV|S}\left(s\right)$ is the IV estimator under point truncation (i.e. the limit of the interval truncated estimator as the interval collapses to a point).
    
    In a homogeneous linear model with normal errors, $ V = \begin{bmatrix} Z & U & T & S & Y \end{bmatrix}' $ will follow a multivariate normal distribution.
    Multivariate normal distributions have the useful property that their conditional distributions have constant covariances across the conditioning level.
    Hence, for all $ V_1, V_2, V_3 \in \{Z, U, T, S, Y\} $ and $ v_0, v_1 \in \mathbb{R} $, we have that
    \[ 
    Cov(V_1, V_2| V_3 = v_0)  = Cov(V_1, V_2| V_3 = v_1). 
    \] 
    It follows that $\beta_{IV|S}\left(s_{0}\right)=\beta_{IV|S}\left(s_{1}\right)$
    for any $s_{0},s_{1}\in\mathbb{R}$.
    Since the stratum specific IV estimator is constant across strata of $ S $, this implies that the IV estimator under adjustment on $ S $ is the same as any stratum specific IV estimator. 
\end{proof}















\chapter*{Instrumental Variables with Treatment-Induced Selection: Exact Bias Results}
\begin{LARGE}Felix Elwert\end{LARGE}\\
\begin{large}University of Wisconsin--Madison\end{large}\vspace{0.3cm}\\
\begin{LARGE}Elan Segarra\end{LARGE}\\
\begin{large}University of Wisconsin--Madison\end{large}\\
\begin{quote} \begin{small}
Instrumental variables (IV) estimation suffers selection bias when the analysis conditions on the treatment. Judea Pearl's [2000:248] early graphical definition of instrumental variables explicitly prohibited conditioning on the treatment. Nonetheless, the practice remains common. 
In this paper, we derive exact analytic expressions for IV selection bias across a range of data-generating models, and for various selection-inducing procedures. 
We present four sets of results for linear models.
First, IV selection bias depends on the conditioning procedure (covariate adjustment vs. sample truncation). Second, IV selection bias due to covariate adjustment is the limiting case of IV selection bias due to sample truncation.
Third, in certain models, the IV and OLS estimators under selection bound the true causal effect in large samples.
Fourth, we characterize situations where IV remains preferred to OLS despite selection on the treatment.
These results broaden the notion of IV selection bias beyond sample truncation, replace prior simulation findings with exact analytic formulas, and enable formal sensitivity analyses.
\end{small} \end{quote}


\section{Introduction}
\input{01_intro.tex}
 
\section{Causal Graphs}\label{sec:causal graphs}
\input{02_causal_graphs.tex}

\section{Instrumental Variables}\label{sec:IV}
\input{03_IV.tex}

\section{Selection Bias in IV: Qualitative Analysis}\label{sec:qualitative}
\input{04_qualitative.tex}

\section{Selection Bias in IV: Quantitative Analysis}\label{sec:quantitative}
\input{05_quantitative.tex}

\section{Conclusion}\label{sec:conclusion}
\input{06_conclusion.tex}

\bibliographystyle{mcp-acm}
