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\title{\textbf{BANA 290 --- Assignment 4}\\[6pt]
       \large Causal Inference in the Smart Campus Incubator:\\
       Instrumental Variables and Regression Discontinuity Design}
\author{Todd Denaro \\ \texttt{denarot@uci.edu}}
\date{April 30, 2026}

\begin{document}
\maketitle

% ─────────────────────────────────────────────────────────────
\section*{1. Instrumental Variables Analysis}
% ─────────────────────────────────────────────────────────────

\subsection*{First-Stage Results}

The first stage regresses AI Intensity (\texttt{AI\_INTENSITY}) on the
instrument, Distance to the Fiber Backbone (\texttt{DISTANCE\_TO\_NODE}).
Table~\ref{tab:first_stage} reports the estimates.

\begin{table}[h!]
\centering
\caption{First-Stage Regression: AI Intensity $\sim$ Distance to Node}
\label{tab:first_stage}
\begin{tabular}{lrrrr}
\toprule
 & Coefficient & Std.\ Error & $t$ & $p$-value \\
\midrule
Intercept           &  67.649 & 2.324 & 29.10 & $<0.001$ \\
Distance to Node (m) & $-0.0189$ & 0.0026 & $-7.29$ & $<0.001$ \\
\midrule
\multicolumn{5}{l}{$F = 53.11$ \quad $R^{2} = 0.431$ \quad $n = 72$} \\
\bottomrule
\end{tabular}
\end{table}

The instrument is \textbf{strong}: the first-stage $F$-statistic of 53.1 far
exceeds the conventional threshold of 10, confirming that distance to the
fiber backbone is a relevant predictor of AI compute access. The negative
coefficient ($\beta = -0.0189$) indicates that each additional metre of
distance reduces weekly AI compute usage by roughly 0.019 hours --- a
meaningful constraint given the range of 150--1485 metres across campus.

\subsection*{OLS vs.\ 2SLS Estimates}

Table~\ref{tab:iv_compare} compares the naive OLS estimate against the
two-stage least squares (2SLS) causal estimate of AI Intensity on Innovation
Score. Standard errors for 2SLS are heteroskedasticity-robust sandwich
estimates using structural residuals (computed from the original endogenous
variable, not the fitted values).

\begin{table}[h!]
\centering
\caption{OLS vs.\ 2SLS: Innovation Score $\sim$ AI Intensity}
\label{tab:iv_compare}
\begin{tabular}{lrrrr}
\toprule
 & Coefficient & Std.\ Error & $t$ & $p$-value \\
\midrule
\textbf{OLS} & 0.3245 & 0.079 & 4.11 & $<0.001$ \\
\textbf{2SLS} & 0.1119 & 0.127 & 0.88 & $0.383$ \\
\bottomrule
\end{tabular}
\end{table}

The OLS estimate ($\hat{\beta}_{\text{OLS}} = 0.325$) is statistically
significant and positive but almost certainly \textbf{upward-biased}. Teams
with stronger inherent ability, deeper experience, or better mentorship
networks tend to both invest more in AI compute \emph{and} score higher on
innovation regardless of infrastructure access. This positive selection
produces a spurious correlation that inflates the OLS coefficient.

The 2SLS estimate ($\hat{\beta}_{\text{2SLS}} = 0.112$) is roughly
one-third the size of the OLS and loses statistical significance
($p = 0.383$). By exploiting only the exogenous variation in AI access
driven by housing-determined distance, 2SLS strips out the confounding
from unobserved team quality. The smaller, noisier estimate suggests that
the raw OLS association was largely driven by selection rather than a
structural causal effect of compute hours on innovation output.

% ─────────────────────────────────────────────────────────────
\section*{2. Regression Discontinuity Design}
% ─────────────────────────────────────────────────────────────

\subsection*{Design and Visual Test}

The RDD exploits the sharp cutoff at an Eligibility Score of 85: teams at
or above 85 automatically received overnight server credits (GPU nights),
while teams below were placed on a mentoring-only waitlist. The running
variable is centred at the cutoff, and local-linear regressions are fit
separately on each side. Figure~\ref{fig:rdd_panels} presents the full
four-panel diagnostic.

\begin{figure}[h!]
\centering
\includegraphics[width=\textwidth]{analysis_results.png}
\caption{IV and RDD Analysis Panels. \textit{Top-left}: First-stage scatter
  with fitted line and $F$-statistic annotation. \textit{Top-right}: OLS
  (orange) vs.\ 2SLS (blue dashed) fitted lines over the raw scatter.
  \textit{Bottom-left}: RDD scatter with local-linear fits on each side of
  the cutoff and the LATE arrow. \textit{Bottom-right}: Density of
  Eligibility Score for the manipulation test.}
\label{fig:rdd_panels}
\end{figure}

Panel~3 (bottom-left) shows a clear visual \textbf{jump} in Innovation Score
at the 85-point threshold. Teams just below the cutoff cluster around scores
of 60--63, while teams just above jump to 68--71, even holding the linear
slope of Eligibility Score constant.

\subsection*{RDD Estimates}

\begin{table}[h!]
\centering
\caption{Sharp RDD Estimates: LATE at Eligibility Cutoff of 85}
\label{tab:rdd}
\begin{tabular}{lrrrr}
\toprule
 & LATE & Std.\ Error & $t$ & $p$-value \\
\midrule
Full sample ($n=72$)        & 8.487 & 0.763 & 11.12 & $<0.001$ \\
Bandwidth $\pm 10$ ($n=68$) & 8.945 & 0.786 & 11.38 & $<0.001$ \\
\bottomrule
\end{tabular}
\end{table}

The Local Average Treatment Effect (LATE) is approximately
\textbf{+8.5 innovation points} at the cutoff, and the estimate is
robust across both the full sample and the narrower $\pm 10$-point
bandwidth (Table~\ref{tab:rdd}). The interaction term
(\texttt{TREATED:SCORE\_C} $= -0.291$, $p = 0.044$) indicates a slightly
flatter slope above the cutoff, consistent with a ceiling effect as teams
approach maximum innovation capacity.

% ─────────────────────────────────────────────────────────────
\section*{3. Assumption Tests}
% ─────────────────────────────────────────────────────────────

\subsection*{Exclusion Restriction (IV)}

The instrument, distance to the fiber backbone, is determined by campus
housing assignment --- a logistical decision made before teams were formed and
entirely independent of the incubator's innovation evaluation. There is no
plausible direct pathway from physical proximity to an underground fiber
cabinet to a team's pitch score. The only channel through which distance
can affect innovation is by constraining AI compute access
(\texttt{AI\_INTENSITY}). This satisfies the exclusion restriction: the
instrument affects the outcome solely through the endogenous regressor.

\subsection*{Continuity / No Manipulation (RDD)}

For the RDD to be valid, teams must not have been able to manipulate their
Eligibility Score to cross the 85-point threshold. Panel~4 (bottom-right of
Figure~\ref{fig:rdd_panels}) plots the density of Eligibility Scores. The
distribution is smooth through the cutoff with no spike just above 85 and
no trough just below. A bin-count check confirms 25 observations in the
$[80, 85)$ window and 19 in the $[85, 90)$ window, yielding a ratio of
0.76 --- well within the range consistent with a continuous density. There
is no evidence of strategic manipulation.

% ─────────────────────────────────────────────────────────────
\section*{4. Interpretation and Policy Implications}
% ─────────────────────────────────────────────────────────────

Both identification strategies converge on the same conclusion: \textbf{AI
compute access causally boosts student innovation}, but the effect is
concentrated in infrastructure access rather than raw compute hours alone.

The IV analysis reveals that the naive OLS coefficient (0.325) is
substantially upward-biased relative to the 2SLS estimate (0.112), driven
by positive selection of high-ability teams into AI-intensive workflows.
Once that selection is purged, the per-hour marginal effect is modest and
statistically imprecise, suggesting diminishing returns to incremental
compute time.

The RDD tells the more policy-relevant story. A team sitting just below the
85-point threshold --- scoring, say, 84.9 --- receives no server credits and
produces an innovation score roughly 8.5 points lower than an otherwise
equivalent team that scored 85.0 and received GPU access overnight. This
discontinuity is large (more than one standard deviation of Innovation Score)
and survives robustness checks. It implies that \textbf{compute access is a
binding constraint at the margin}: teams are not resource-constrained on
mentorship or ideas but on the raw infrastructure needed to prototype and
iterate AI-driven products.

The policy implication is clear. Lowering the server-credit threshold,
introducing a tiered compute allocation that extends partial credits to
teams in the 75--85 range, or expanding campus fiber capacity to reduce
average distances would all generate measurable innovation returns. The
marginal cost of providing one team an extra 8 GPU nights is low relative
to an 8.5-point gain in innovation output --- a compelling return on
infrastructure investment.

\end{document}