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Accelerating the Adoption of Residential Solar Power Systems: Policy Analysis using a Dynamic Structural Model
\begingroup
\hoffset=0pt \voffset=0pt\relax
\includepdf[pages=-,fitpaper=true,noautoscale=true]{DDCM_paper_06-30-2026.pdf}
\endgroup
\RUNTITLE{Accelerating Residential Solar Adoption}
\TITLE{Accelerating the Adoption of Residential Solar Power Systems: Policy Analysis using a Dynamic Structural Model}
\ARTICLEAUTHORS{
\AUTHOR{Sebastian Souyris}
\AFF{Lally School of Business, Rensselaer Polytechnic Institute, NY 12180,
\EMAIL{[email removed]}}
\AUTHOR{Jason Duan, Anant Balakrishnan}
\AFF{McCombs School of Business, The University of Texas at Austin, Austin, TX 78712, \EMAIL{\{[email removed], [email removed]\}}}
\AUTHOR{Varun Rai}
\AFF{LBJ School of Public Affairs, The University of Texas at Austin, Austin, TX 78712, \EMAIL{[email removed]}}
}
\ABSTRACT{}
\KEYWORDS{renewable energy, diffusion, dynamic discrete choice model, Bayesian statistics} \HISTORY{.}
\ECSwitch
\noindent\textbf{Accelerating the Adoption of Residential Solar Power Systems: Policy Analysis using a Dynamic Structural Model}\\
\noindent\textbf{Electronic Companion}\\
\raggedbottom
\setcounter{equation}{0}
\begin{APPENDICES}
\section{Model Identification and Rebate Exogeneity}\label{sec:ECidentification}
\subsection{Identification}
The observed data include each household's choice $a_{it}$ in every
period, the economic and geographic segment of household $i$, the
state variables $c_{t}$ and $h_{it}$, and the net present value
$NPV_{it}(c_{t})$. The parameters to be identified are $\theta_{i}=\left\{ \rho,\alpha_{e_{i}},\gamma_{g_{i}},\xi_{i}\right\} $.
Here, we discuss in detail how the parameters in the utility $\theta_{i}=\left\{ \rho,\alpha_{e_{i}},\gamma_{g_{i}},\xi_{i}\right\} $
and the variance $\sigma_{\xi}^{2}$ of the household-specific
unobserved heterogeneity term can be identified from the data.
The state transition functions $p\left(c_{t+1}|c_{t}\right),\text{ }p\left(h_{i,t+1}|h_{it}\right)$
are identified from the observed costs and photovoltaic (PV) installations over time.
For each economic and geographic segment, we observed a large number
(thousands) of households. Hence, for each segment, $p\left(a_{it}|c_{t},h_{it}\right),a_{it}=0,1$
is identifiable semiparametrically (e.g., by a logistic regression
of $a_{it}$ on the polynomials of $c_{t}$ and $h_{it}$). $p\left(a_{it}|c_{t},h_{it}\right)$
is assumed to be the outcome of the parametric structural model, i.e., $p\left(a_{it}|c_{t},h_{it},\theta_{i}\right)=p\left(a_{it}|c_{t},h_{it}\right)$.
Similar to \citet{Hotz1993}, we have
\begin{eqnarray}
\log p\left(a_{it}=1|c_{t},h_{it},\theta_{i}\right)-\log p\left(a_{it}=0|c_{t},h_{it},\theta_{i}\right) & = & \bar{u}_{i1t}(c_{t},h_{it},\theta_{i})-\beta EV_{i}\left(c_{t},h_{it},a_{it}=0,\theta_{i}\right)\nonumber \\
\text{where \ensuremath{\bar{u}_{i1t}}(\ensuremath{c_{t}},\ensuremath{h_{it}},\ensuremath{\theta_{i}})} & = & \rho+\alpha_{e_{i}}NPV_{it}(c_{t})+\gamma_{g_{i}}h_{it}+\xi_{i}.\label{eq:A4}
\end{eqnarray}
Because the discount factor $\beta$ in a dynamic discrete choice
model is generally not separately identifiable from the structural
parameters \citep{Magnac2002}, we fix its value. Next, we discuss
how $\theta_{i}=\left\{ \rho,\alpha_{e_{i}},\gamma_{g_{i}},\xi_{i}\right\} $
and $\sigma_{\xi}^{2}$ are identified. In general, by Corollary~3
in \citet{Magnac2002}, the dynamic discrete choice model parameters
are uniquely identified given fixed $\beta$ and the state transition
probabilities (densities), $p\left(c_{t+1}|c_{t}\right),\text{ }p\left(h_{i,t+1}|h_{it}\right),\text{ }p\left(\varepsilon_{iat}\right)$.
We then focus on how the specific parameters in our model are identified.
From equation \eqref{eq:A4}, if $EV_{i}\left(c_{t},h_{it},a_{it}=0,\theta_{i}\right)$
is known, then, for each segment, $\rho,\alpha_{e_{i}},\gamma_{g_{i}}$ are
identified by a linear regression of the log-choice-probability difference
in equation \eqref{eq:A4} adjusted for expected value function, given
that $p\left(a_{it}|c_{t},h_{it},\theta_{i}\right)=p\left(a_{it}|c_{t},h_{it}\right)$
has been consistently estimated. The household-specific $\xi_{i}$
is estimated by the residuals of the regression and $\sigma_{\xi}^{2}$
is identified by the variance of the residuals across all segments.
In practice, $EV_{i}\left(c_{t},h_{it},a_{it}=0,\theta_{i}\right)$
is unknown. Given a candidate value in the estimation iterations,
$\theta_{i}=\left\{ \rho,\alpha_{e_{i}},\gamma_{g_{i}},\xi_{i}\right\} $,
$EV_{i}\left(c_{t},h_{it},a_{it}=0,\theta_{i}\right)$ is the fixed-point
solution of the expected-value recursion in Appendix~C (equation \eqref{eq:emax_approx}). The identification and
estimation therefore follow the nested fixed-point logic of \citet{Rust1987a}. In the subsequent iteration, holding fixed $EV_{i}\left(c_{t},h_{it},a_{it}=0,\theta_{i}\right)$
from the previous step, $\theta_{i}=\left\{ \rho,\alpha_{e_{i}},\gamma_{g_{i}},\xi_{i}\right\} $
is estimated again from equation \eqref{eq:A4}. This iterative procedure
converges to the fixed point of the model \citep{Rust1987a,Magnac2002}, yielding the final estimates of $\theta_{i}=\left\{ \rho,\alpha_{e_{i}},\gamma_{g_{i}},\xi_{i}\right\} $.
\subsection{Institutional Background and Exogeneity Evidence}\label{sec:ECinstitutional}
The net cost that drives adoption in our model is determined in part
by the residential PV rebate, and the institutional record
shows that Austin Energy set this rebate through an administrative
budget process rather than in response to individual adoption decisions.
Austin Energy funded the rebate from a fixed annual budget approved
in the City of Austin budget cycle and disbursed it as a posted dollars-per-watt
rate that applied uniformly to every qualifying applicant. That is,
it did not negotiate the subsidy household by household or condition
it on a neighborhood's adoption history. The same administratively recorded
schedule is described in prior publications \citep{Rai2015,Rai2013}.
When the installation-cost benchmarks used in the budgeting process
changed, Austin Energy revised the posted rate administratively and
announced the new rate publicly for the next fiscal year. The other
components of the net cost include the federal investment tax credit
by statute \citep{EPAct2005,EESA2008} and the Value-of-Solar tariff,
which replaced residential net metering on October~1, 2012, through
a separate citywide proceeding \citep{AustinEnergyVOS2017,DSIRE5669}.
The observed posted rebate rates in our data display a small number
of discrete downward revisions over the time window. The program started
in 2004 at \$5.00 per watt and then held near \$4.50 through fiscal
years 2005--2008 \citep{SolarAustin2011}. At the beginning of fiscal
2009 the rate fell to \$3.75. It was then cut to \$2.00, effective
immediately, on June~11, 2012, and to about \$1.50 in mid-2013.
Two features of this process support treating the net cost as exogenous
to the individual adoption decision. First, the rate changes were
infrequent, discrete, and downward-only, and each was triggered by
a documented decline in installation cost (a 17\% decline in 2010
and an 11\% decline over the first half of 2011), rather than by any
household's characteristics or by local adoption \citep{SolarAustin2011}.
Second, the rate was posted and applied uniformly, so an individual
applicant could neither bargain over the subsidy nor reliably time
it.
One remaining concern is that Austin Energy may set the annual rebate
budget and rate in anticipation of aggregate demand shocks in the
following year. To further assess whether the rebate rate could be
endogenous and bias our estimates, we conduct an empirical test of
whether household net present value $\text{NPV}_{it}$ has different effects
on PV adoption depending on whether we include period-specific temporal
fixed effects that control for unobserved demand shocks over time
\citep[Chapter~10]{Wooldridge2010}. In our model, $\text{NPV}_{it}$ is the
only variable that is a function of the rebate rate. Because the dependent
variable of installation is a quarterly dummy variable over time, we
use the following reduced discrete hazard model
\[
G\left(\lambda_{it}\right)=\xi_{z}+\eta_{t}+\beta_{1}\text{NPV}_{it}+\beta_{2}H_{it}+\beta_{3}X_{i},
\]
where the hazard rate $\lambda_{it}$ of installation is modeled by
the complementary log-log ($\log[-\log[1-\lambda_{it}]]$) function.
Here, $\eta_{t}$ is the fixed effect that captures the unobserved
aggregate temporal demand shocks for all households in the city, $\xi_{z}$
is the zip-code fixed effect, $X_{i}$ represents household characteristics,
and $H_{it}$ is the number of neighborhood installations within the
one-mile radial ring. If the rebate rate is endogenous because of
the anticipation of aggregate demand shocks, we should expect $\text{NPV}_{it}$
to be correlated with $\eta_{t}$. Therefore, the model that omits $\eta_{t}$,
\[
G\left(\lambda_{it}\right)=\xi_{z}+\beta_{1}\text{NPV}_{it}+\beta_{2}H_{it}+\beta_{3}X_{i},
\]
should produce a materially different estimate of the coefficient $\beta_{1}$.
However, our estimation of these two models yields estimates of $\beta_{1}$
that differ by less than one standard error (Table~\ref{tab:ECexogeneity}), which indicates that the
variation in $\text{NPV}_{it}$ identifying the price response is orthogonal to
the citywide temporal shocks captured by $\eta_{t}$.
\begin{table}[H]
\begin{center}
\caption{Sensitivity of the net-present-value coefficient $\beta_{1}$ to temporal fixed effects in the discrete hazard model.\label{tab:ECexogeneity}}
{\scriptsize
\begin{tabular}{lrrr}
\toprule
$\text{NPV}_{it}$ coefficient & Estimate & S.E. & p-value\bigstrut\\
\midrule
Model with temporal fixed effects & $2.36\times10^{-6}$ & $0.085\times10^{-6}$ & $\simeq0$\\
Model without temporal fixed effects & $2.31\times10^{-6}$ & $0.072\times10^{-6}$ & $\simeq0$\\
\bottomrule
\end{tabular}
}
\end{center}
\end{table}
\clearpage
\section{Proof of Consistency of Using Cluster-Based Sampling}
\label{sec:Proofclusters} In Section 4.2, we aggregate households
into clusters with similar characteristics to reduce the computational
cost of the estimation. Let the total sample size of households in
a city be $M.$ These households are partitioned into $K$ clusters,
each cluster is denoted as the set $M_{k}$, whose size is $m_{k}$.
Observe that, if the data of a household used in the likelihood-based
estimation is not from any cluster, it can be viewed as in a cluster
of size one by itself. Therefore, all the households can be viewed
as in a cluster $M_{k},k=1,...,K$. We denote household $i$'s choice
as $a_{it}\in\{0,1\}$. Notice that the cluster size $m_{k(i)},i\in M_{k}$
depends on the choice $a_{it}$. If $a_{it}=1$ for some $t=1,...,T$,
then the size $m_{k(i)}=1$. If $a_{it}=0$ for all $t=1,...,T$,
then the cluster size $m_{k(i)}$ can be greater than one. We fix the cluster composition throughout the observation window, so the cluster sizes do not depend on time $t$.
Each household has a set of characteristic variables $X_{it}$ including
economic segment, geographic segment and NPV. We consider $X_{it}$'s
as random variables independent across households, i.e., $X_{it}=\left[X_{i1},...,X_{iT}\right]\overset{iid}{\sim}p\left(x_{1},...,x_{T}\right)$,
which is the distribution of household characteristics in the population.
The function $p\left(x_{1},...,x_{T}\right)$ can represent a pdf
if $x_{t}$ is continuous or a pmf if $x_{t}$ is discrete. (The corresponding
$\intop p\left(x_{1},...,x_{T}\right)dx_{1}\cdots dx_{T}$ is integrated
with respect to either a Lebesgue measure or a counting measure).
The choice probability defined by Equation (13) of the paper can be simplified
as $p_{iat}=p\left(a|x_{it};\vartheta\right),a=0,1,$ where $\vartheta$
represents the set of model parameters that we are interested in estimating.
The full log-likelihood function if we use the entire sample in the
estimation is
\begin{equation}
L^{a}=\sum_{i=1}^{M}\left\{ \sum_{t=1}^{T}\log p\left(a_{it}|x_{it};\vartheta\right)\right\}.
\end{equation}
If we employ the cluster-based sampling method above and randomly
choose one household from each cluster, the log-likelihood based on
our cluster-based sampling method is
\begin{equation}
L^{c}=\sum_{k=1}^{K}\left\{ m_{k(i)}\sum_{t=1}^{T}\log p\left(a_{it}|x_{it};\vartheta\right)\right\} ,\label{eq:cluster-loglkhd}
\end{equation}
where $k\left(i\right)$ indicates the cluster $k$ from which household
$i$ is selected at random ($i\sim \text{Unif}(M_k)$).
\begin{proposition}
Assume the probabilistic choice model described in Section 4.1 and
the cluster-based sampling method described above. Let $L^{a}$ and
$L^{c}$ designate the full and the weighted cluster-based sampling
log-likelihood functions. Assume further that the probabilistic choice $p\left(a_{it}|x_{it};\vartheta\right)$,
the structure of the attribute space, and the attribute distribution
$p\left(x_{1},...,x_{T}\right)$ imply a unique maximum $\vartheta^{*}$ when
the sample size $M\rightarrow\infty$. Then the maximum likelihood
estimators of $L^{a}$ and $L^{c}$ converge to the same $\vartheta^{*}$
almost surely when the sample size $M\rightarrow\infty$ and the number
of clusters $K\rightarrow\infty$.
\end{proposition}
\proof{Proof}
When the cluster-based sampling method is applied, the clusters sizes
form a distribution $p\left(m_{k(i)}|a_{i1},...,a_{iT}\right)$. The
conditional distribution of household characteristics given clustering
is $p\left(x_{i1},...,x_{iT}|m_{k(i)},a_{i1},...,a_{iT}\right)$.
Notice that the random variables in the full log-likelihood $L^{a}$
and $L^{c}$ have different distributions. In the full log-likelihood
$L^{a}$, $a_{it}$ and $x_{it}$ have simply the joint distribution
\begin{equation}
p_{i}\left(a_{i1},...,a_{iT},x_{it},...,x_{iT}\right)=\left[\prod_{t=1}^{T}p\left(a_{it}|x_{it};\vartheta\right)\right]p\left(x_{it},...,x_{iT}\right),
\end{equation}
whereas in $L^{c}$, the random variables $a_{it}$, $m_{k(i)}$ and
$x_{it}$ of household $i$ have a different joint distribution because
of cluster-based sampling. We have
\begin{eqnarray}
& & p\left(a_{i1},...,a_{iT},x_{it},...,x_{i(k)T},m_{k}\right)\nonumber \\
& & =\frac{1}{m_{k(i)}}p\left(x_{i1},...,x_{iT}|m_{k(i)}\right)p\left(m_{k(i)}|a_{i1},...,a_{iT}\right)p\left(a_{i1},...,a_{iT}\right),
\end{eqnarray}
where
\[
p\left(a_{i1},...,a_{iT}\right)=\int p\left(a_{i1},...,a_{iT},x_{it},...,x_{iT}\right)dx_{i1}\cdots dx_{iT},
\]
and $1/m_{k(i)}$ reflects the probability that $i$ is randomly chosen
from cluster $M_{k}$ ($m_{k(i)}=1$ when $i$ is in a cluster of
size one, which implies it will always be chosen if it is not in a
cluster with more than one household).
Let us consider the log-likelihood $L^{a}$ divided by $M$:
\[
D_{M}^{a}\left(a,x,\vartheta\right)=\frac{1}{M}\sum_{i=1}^{M}\left\{ \sum_{t=1}^{T}\log p\left(a_{it}|x_{it};\vartheta\right)\right\} .
\]
When $M\rightarrow\infty$, by the strong law of large numbers\footnote{Here, $\int_{x_{t}}dx_{1}\cdots dx_{T}$ is an abbreviation of $\int_{x_{1}}\cdots\int_{x_{T}}dx_{1}\cdots dx_{T}$
and $\underset{a_{t}=\left\{ 0,1\right\} }{\sum}$ is for $\underset{a_{1}=\left\{ 0,1\right\} }{\sum}\underset{a_{2}=\left\{ 0,1\right\} }{\sum}\cdots\underset{a_{T}=\left\{ 0,1\right\} }{\sum}$.},
\begin{eqnarray*}
& & D_{M}^{a}\left(a,x,\vartheta\right)\overset{a.s.}{\rightarrow}D^{a}\left(\vartheta\right)\\
& & =\int_{x_{t}}\sum_{a_{t}=\left\{ 0,1\right\} }\left\{ \sum_{t=1}^{T}\log p\left(a_{t}|x_{t};\vartheta\right)\right\} \prod_{t=1}^{T}p\left(a_{t}|x_{t};\vartheta\right)p\left(x_{1},...,x_{T}\right)dx_{1}\cdots dx_{T}.
\end{eqnarray*}
Now, let us consider the log-likelihood $L^{c}$ divided by $K$:
\[
D_{K}^{c}\left(a,x,\vartheta\right)=\frac{L^{c}}{K}=\frac{1}{K}\sum_{k=1}^{K}\left\{ m_{k(i)}\sum_{t=1}^{T}\log p\left(a_{it}|x_{it};\vartheta\right)\right\} .
\]
When $M\rightarrow\infty$, because the cluster sizes $m_{k}$ do not
increase with $M$, the number of clusters $K\rightarrow\infty$.
Hence, by the strong law of large numbers,
\begin{eqnarray*}
& & D_{K}^{c}\left(a,x,\vartheta\right)\overset{a.s.}{\rightarrow}D^{c}\left(\vartheta\right)=\int_{x_{t}}\int_{m_{k}}\sum_{a_{t}=\left\{ 0,1\right\} }\left\{ m_{k}\sum_{t=1}^{T}\log p\left(a_{t}|x_{t};\vartheta\right)\right\} \cdot\\
& & \frac{1}{m_{k}}p\left(x_{1},...,x_{T}|m_{k}\right)p\left(m_{k}|a_{1},...,a_{T}\right)p\left(a_{1},...,a_{T}\right)dm_{k}dx_{1}\cdots dx_{T}\\
& & =\int_{x_{t}}\int_{m_{k}}\sum_{a_{t}=\left\{ 0,1\right\} }\left\{ \sum_{t=1}^{T}\log p\left(a_{t}|x_{t};\vartheta\right)\right\} p\left(x_{1},...,x_{T},m_{k},a_{1},...,a_{T}\right)dm_{k}dx_{1}\cdots dx_{T}\\
& & =\int_{x_{t}}\sum_{a_{t}=\left\{ 0,1\right\} }\left\{ \sum_{t=1}^{T}\log p\left(a_{t}|x_{t};\vartheta\right)\right\} p\left(x_{1},...,x_{T},a_{1},...,a_{T}\right)dx_{1}\cdots dx_{T}\\
& & =\int_{x_{t}}\sum_{a_{t}=\left\{ 0,1\right\} }\left\{ \sum_{t=1}^{T}\log p\left(a_{t}|x_{t};\vartheta\right)\right\} \prod_{t=1}^{T}p\left(a_{t}|x_{t};\vartheta\right)p\left(x_{1},...,x_{T}\right)dx_{1}\cdots dx_{T}.
\end{eqnarray*}
Therefore, when $M\rightarrow\infty$, $D^{c}\left(\vartheta\right)=D^{a}\left(\vartheta\right)$,
so $D_{K}^{c}\left(a,x,\vartheta\right)$ and $D_{M}^{a}\left(a,x,\vartheta\right)$
are asymptotically equivalent. Observe that
\begin{eqnarray*}
\int_{x_{t}}\sum_{a_{t}=\left\{ 0,1\right\} }\prod_{t=1}^{T}p\left(a_{t}|x_{t};\vartheta\right)p\left(x_{1},...,x_{T}\right)dx_{1}\cdots dx_{T} & = & 1\text{ and}\\
\sum_{t=1}^{T}\log p\left(a_{t}|x_{t};\vartheta\right) & = & \log\prod_{t=1}^{T}p\left(a_{t}|x_{t};\vartheta\right)
\end{eqnarray*}
Hence, by Lemma 1 in \cite{Manskietal1977} and \cite{Rao1973}
$D^{c}\left(\vartheta\right)$ and $D^{a}\left(\vartheta\right)$
are maximized at the same and unique $\vartheta=\vartheta^{*}.$ And
by Lemma 2 of \cite{Manskietal1977}, both $\hat{\vartheta}_{M}$,
which maximizes $D_{M}^{a}\left(a,x,\vartheta\right)$, and $\hat{\vartheta}_{K}$,
which maximizes $D_{K}^{c}\left(a,x,\vartheta\right)$, converge to
the same $\vartheta^{*}$ almost surely.{\hfill \ensuremath{\Box}}\endproof
~\\
\noindent\textbf{Application to the dynamic structural model estimation}
We aggregate non-adopting households into clusters with similar characteristics within each census block group (www2.census.gov) using the k-means clustering technique, with home market value, house size, tree cover, and received irradiance as the clustering features.
Figure \ref{fig:aggbkgexample} shows the households before and after the clustering. Figures \ref{fig:aggbkgexample}(a) and (b) illustrate the Austin Energy service area, where the red dots represent adopters during the analysis period (2004-2013) and each black dot is a potential adopter in Figure \ref{fig:aggbkgexample}(a) and a cluster of potential adopters in Figure \ref{fig:aggbkgexample}(b).
Figure \ref{fig:aggbkgexample}(c) shows magnified plots of two clusters within a particular census block group, where the black dots represent potential adopters and the two green dots mark the centroids of these two clusters. The clusters remain constant over the time frame of our observations.
The clustering yields 1,185 clusters of size greater than one, with an average cluster size of 26 and a median of 7. Given this large number of clusters, the asymptotic result above applies, and the clustering does not introduce appreciable bias into our estimation.
\begin{figure}[H]
{ \centering
\begin{tabular}{C{5cm} C{5cm} C{7cm}}
{\footnotesize(a) Non-adopters disaggregated}&{\footnotesize(b) Non-adopters in clusters} & {\footnotesize(c) Non-adopters disaggregated and cluster centroids}\\
\includegraphics[width=1\linewidth]{sp_NonAndAdopters.pdf} &
\includegraphics[width=1\linewidth]{sp_ClustersAndAdopters.pdf} &
\includegraphics[width=1\linewidth]{aggBkgExample.pdf}\\
\end{tabular}
\caption{PV current adopters and potential future adopters in 2013, Austin, TX.}
\label{fig:aggbkgexample}
}
{\footnotesize
\setlength{\baselineskip}{9pt}
\noindent \emph{Note.} Red dots are adopters. Black dots are non-adopters. Green dots mark the cluster centroids.
\par}
\end{figure}
\clearpage
\section{Estimation and Simulation Algorithms}
We reproduce in this appendix the equations presented in the paper so that the algorithm's equation references are self-contained. See the paper for the detailed explanations.
\vspace{0.5cm}
\begin{minipage}{\textwidth}
\subsubsection*{Dynamic discrete choice model notation}
\begin{center}
\vskip 10pt
{\footnotesize
\begin{tabular} {p{1.3 in} p{4.5 in}}
\toprule
\multicolumn{2}{l}{\textbf{Indices and Sets}}\bigstrut\\
\midrule
$i \in M$ & Set of potential households adopters.\\
$e \in E$ & Set of economic segments.\\
$g\in G$ & Set of geographic segments.\\
$t \in \{1,\ldots, T\}$ & Periods (quarters). \\
$d_l \in D$ & Set of discrete distances to define neighborhoods by distance radius rings. $D = \{d_1<\ldots< d_{|D|}\}$ and $d_0 = 0$.\\
\midrule
\multicolumn{2}{l}{\textbf{Input Data}}\bigstrut\\
\midrule
$w_{i}$ & PV system size of household $i$.\\
$q_{it}$ & Expected electricity generated per watt by household $i$'s PV system in period $t$.\\
$v_t$ & Bill credits (i.e., solar energy rate) for the electricity produced by PV in period $t$.\\
$p_t$ & PV installation cost in period $t$.\\
$r_{t}$ & PV rebate available in period $t$.\\
$\text{ITC}_{t}$ & Federal Investment tax credit in period $t$.\\
$e_i \in E$ & Economic segment of household $i$.\\
$g_i \in G$ & Geographic segment of household $i$.\\
$t_i$ & Period of household $i$'s adoption decision. The PV system is installed in the following period. And $t_i = \infty$ implies $i$ does not adopt.\\
$\tau_i$ & Period of property $i$'s built. If the property is built before the beginning of the estimation window, then $\tau_i = 1$.\\
${M}_t$ & Set of households that have adopted before period $t$. ${M}_t = \{i \in M: t_i < t\}$.\\
${M}^c_t$ & Set of households that have not adopted in period $t$. ${M}^c_t = {M} \setminus {M}_t$.\\
$\beta$& Household discount factor.\\
$L$ & Lifespan of a PV system (80 quarters).\\
\midrule
\multicolumn{2}{l}{\textbf{Decision variable}}\bigstrut\\
\midrule
$a_{it} \in \{0,1\}$& Household $i$'s adoption decision in period $t$; i.e., it equals 1 if household $i$ adopts and 0 otherwise.\\
\midrule
\multicolumn{2}{l}{\textbf{State variables}}\bigstrut\\
\midrule
$c_{t}$ & Net cost per watt of PV in period $t$. $c_{t} = (p_t - r_t)(1- \textrm{ITC}_t)$.\\
$\bm{h}_{it} = \{h_{i1t}, \ldots, h_{i|D|t}\}$ & Previous installations within radial distance $d\in (d_{l-1}, d_l]$ of household $i$, $l = 1,\ldots,|D|$.\\
$\varepsilon_{iat}$ & Additive random shock in household $i$'s utility in period $t$.\\
\midrule
\multicolumn{2}{l}{\textbf{Structural parameters}}\bigstrut\\
\midrule
$\rho$ & Constant intercept of the utility function.\\
$\alpha_{e}$ & Weight of solar economic benefit for economic segment $e \in E$.\\
$\bm{\gamma}_{g} = \{\gamma_{g1}, \ldots, \gamma_{gD}\}$ & Neighborhood effects of previous installations within radial distance $d\in (d_{l-1}, d_l]$, $l = 1,\ldots,|D|$, for geographic segment $g \in G$.\\
$\bm\theta = \{\rho, \bm\alpha, \bm{\gamma}\}$ & Vector of structural parameters.\\
$\xi_{i}$ & Household unobserved heterogeneity.\\
$\sigma_\xi$ & Standard deviation of household heterogeneity $\xi_i$.\\
\bottomrule
\end{tabular}
}
\end{center}
\end{minipage}
\clearpage
\subsubsection*{Per-Period Utility for PV Adoption}
\begin{description}
\item[]Utility for adopting:
\begin{align}
&& u_{i1t}(c_t,\bmh_{it},\xi_{i},\varepsilon_{i1t};\rho, \alpha_{e_i},\bm\gamma_{g_i})&= \rho + \alpha_{e_i} \text{NPV}_{it}(c_t) +\bm\gamma_{g_i}'\bmh_{it} + \xi_{i}+\varepsilon_{i1t},\label{eq:ui1t}\\
\text{where}& && \nonumber\\
& &\text{NPV}_{it}(c_t)&=
\left(\sum\limits_{\tau = t+1}^{t+1+L} \beta^{\tau-t}v_\tauq_{i\tau} - c_t\right)w_i. \label{eq:vpnit}
\end{align}
\item[]Utility for not adopting:
\begin{align}
u_{i0t}(\varepsilon_{i0t}) &= \varepsilon_{i0t}.\label{eq:ui0t}
\end{align}
\end{description}
\subsubsection*{Dynamic Optimal Decision of Adoption}
\begin{align}
V_{it}(c_t,\bmh_{it},\xi_{i} ,\bm{\varepsilon}_{it}; \bm\theta) & = \max\limits_{a \in \{0,1\}}\mathcal{V}_{iat} (c_t,\bmh_{it}, \xi_{i} ,\bm{\varepsilon}_{it}; \bm\theta),\label{eq:vit}
\end{align}
where $\mathcal{V}_{iat}$ is the choice specific value function for making the decision $a \in \{0,1\}$:
\begin{align}
\mathcal{V}_{i0t} (c_t,\bmh_{it}, \xi_{i} ,\bm{\varepsilon}_{it}; \bm\theta)& =
u_{i0t}(\varepsilon_{i0t}) + \beta E_{c, \bmh,\bm{\varepsilon}}\left[V_{it+1}(c, \bmh,\xi_i,\bm{\varepsilon}; \bm\theta)|c_t,\bmh_{it},\bm{\varepsilon}_{it}\right]\nonumber\\
&= \varepsilon_{i0t} + \beta E_{c,\bmh ,\bm{\varepsilon}_i}\left[V_{it+1}(c, \bmh,\xi ,\bm{\varepsilon}; \bm\theta)|c_t,\bmh_{it},\bm{\varepsilon}_{it}\right].
\label{eq:vi0t}\\
\mathcal{V}_{i1t} (c_t,\bmh_{it}, \xi_{i} ,\bm{\varepsilon}_{it}; \bm\theta)& = u_{i1t}(c_t,\bmh_{it},\xi_{i},\varepsilon_{i1t} ; \bm\theta) \nonumber\\
&= \rho + \alpha_{e_i} \text{NPV}_{it}(c_t) +\bm\gamma_{g_i}'\bmh_{it} + \xi_{i}+\varepsilon_{i1t}.\label{eq:vi1t}
\end{align}
Household $i$ will adopt at $t$ if $\mathcal{V}_{i1t} (c_t,\bmh_{it}, \xi_{i} ,\bm{\varepsilon}_{it}; \bm\theta) \geq \mathcal{V}_{i0t} (c_t,\bmh_{it}, \xi_{i} ,\bm{\varepsilon}_{it}; \bm\theta)$.
\subsubsection*{State Variables, Their Transition Equations and Other Model Specifications}
\begin{align}
c_{t+1} &= \kappa c_{t} + \epsilon^{c}_t, & \epsilon^{c}_t &\sim N(0,\sigma^2_{c}), \label{eq:ct}\\
h_{i,t+1} &= h_{it} + \Delta_{it}, & \Delta_{it} &\sim \text{Exponential}\left(\tfrac{1}{(\lambda_i-1)h_{it}}\right), \quad \lambda_i>1,\label{eq:ht}\\
\xi_{i} &\sim N(0,\sigma^2_{\xi}), & \sigma^2_{\xi} &\sim InverseGamma(\pi_1,\pi_2). \label{eq:xii}
\end{align}
\subsection*{Bayesian Estimation}
\vskip 10pt
\begin{tabular} {p{1.3 in} p{4.5 in}}
$r = 1,\ldots R$ & Iterations of the Metropolis–within–Gibbs method.\\
$\left\{c_t^{j(r)}\right\}_{j=1}^J$ & Random grid of size $J$ of the net cost state variable $c_t$ in iteration $r$ using \eqref{eq:ct}.\\
$f(c_{t+1}| c_t)$ & Gaussian Kernel resulting from \eqref{eq:ct}.\\
$K_\eta\left(\bm\theta^{(n)} - \bm\theta^{(r)} \right)$ & Multivariate Gaussian kernel with bandwidth $\eta > 0$.\\
\end{tabular}
\begin{align}
\hat E\Big[\hat V^{(r)}_{it+1}\left(c, \bm{h}_{it+1},\xi ; \bm\theta\right)&|c^{j(r)}_{t},\bm{h}_{it},\xi^{(r)}_{i}; \bm\theta^{(r)}\Big] \nonumber\\
&= \frac{\sum_{n = r-N}^{r-1} K_\eta\left(\bm\theta^{(n)} - \bm\theta^{(r)} \right) \sum_{j=1}^J \hat V^{(n)}_{it+1}\left({c}^{j(n)}_{t+1},\bm{h}_{it+1},\xi^{(n)}_{i} ; \bm\theta^{(n)}\right)f\left({c}^{j(n)}_{t+1}|{c}^{j(r)}_{t}\right)}{\sum_{n = r-N}^{r-1} K_\eta\left(\bm\theta^{(n)} - \bm\theta^{(r)} \right)\sum_{j=1}^J f\left({c}^{j(n)}_{t+1}|{c}^{j(r)}_{t}\right)}, \label{eq:emax_approx}
\end{align}
where
\begin{align}
\hat V^{(n)}_{it}({c}^{j(n)}_{t},\bm{h}_{it},\xi^{(n)}_{i} ; \bm\theta^{(n)}) &
= \log\left(\exp\hat{\mathcal{V}}^{(n)}_{i0t}\left({c}^{j(n)}_{t},\bm{h}_{it},\xi^{(n)}_{i}; \bm\theta^{(n)}\right)+ \exp\hat{\mathcal{V}}^{(n)}_{i1t}\left({c}^{j(n)}_{t},\bm{h}_{it},\xi^{(n)}_{i}; \bm\theta^{(n)}\right)\right), \label{eq:pseudo_vf}
\end{align}
and
\begin{align}
\hat{\mathcal{V}}^{(n)}_{i0t}\left({c}^{j(n)}_{t},\bm{h}_{it},\xi^{(n)}_{i} ; \bm\theta^{(n)}\right)&
= \beta \hat E\left[\hat V^{(n)}_{it+1}\left({c}, \bm{h}_{it+1},\xi; \bm\theta\right)|{c}^{j(n)}_{t},\bm{h}_{it},\xi^{(n)}_{i}; \bm\theta^{(n)}\right]\\
\hat{\mathcal{V}}^{(n)}_{i1t}\left({c}^{j(n)}_{t},\bm{h}_{it},\xi^{(n)}_{i} ; \bm\theta^{(n)}\right)&
=\rho^{(n)} + \alpha^{(n)}_{e_i} \textrm{NPV}_{it}\left({c}^{j(n)}_t\right) +\bm\gamma^{(n)'}_{g_i}\bm{h}_{it} + \xi^{(n)}_{i}.
\end{align}
Then, at iteration $r$, the probability of adopting at period $t$ is:
\begin{align}
\hat p^{(r)}_{i1t}(c_t,\bm{h}_{it}, \xi^{(r)}_{i};\bm\theta^{(r)}) &= \frac{\exp\hat{\mathcal{V}}^{(r)}_{i1t} \left({c}_{t},\bm{h}_{it},\xi^{(r)}_{i}; \bm\theta^{(r)}\right)}{\exp\hat{\mathcal{V}}^{(r)}_{i1t}\left({c}_{t},\bm{h}_{it},\xi^{(r)}_{i} ; \bm\theta^{(r)}\right) +\exp\hat{\mathcal{V}}^{(r)}_{i0t}\left({c}_{t},\bm{h}_{it},\xi^{(r)}_{i} ; \bm\theta^{(r)}\right)}.
\end{align}
Recall that $\tau_i$ is the period in which household $i$'s property is built ($\tau_i=1$ if built before or at the first period) and $t_i$ is the adoption period ($t_i=T$ if $i$ does not adopt during the horizon). Then, at iteration $r$, household $i$'s log-likelihood is:
\begin{align}
l_i^{(r)}&\left(\bm{c},\bm{h}_{i}, \xi^{(r)}_{i};\bm{\theta}^{(r)}\right) \nonumber\\
&= \sum_{t=\tau_i}^{t_i}\left\{1_{\{a_{it}=1\}}\log\left({\hat{p}}_{i1t}^{(r)}\left(c_t,\bm{h}_{it}, \xi^{(r)}_{i};\bm{\theta}^{(r)}\right)\right)+1_{\{a_{it}=0\}}\log\left(1-{\hat{p}}_{i1t}^{(r)}\left(c_t,\bm{h}_{it}, \xi^{(r)}_{i};\bm{\theta}^{(r)}\right)\right)\right\},\nonumber
\end{align}
And the full likelihood is:
\begin{align}
\mathcal{L}^{(r)}\left(\bm{c},\bm{h}, \bm\xi^{(r)};\bm{\theta}^{(r)}\right) &= \sum_{i \in M} l^{(r)}_i\left(\bm{c},\bm{h}_{i}, \xi^{(r)}_{i};\bm{\theta}^{(r)}\right).\label{eq:loglik}
\end{align}
\begin{center}
\textbf{Markov Chain Monte Carlo (MCMC) Estimation Algorithm}
{
{\scriptsize
\begin{algorithm}[H]
\SetAlgoLined
\DontPrintSemicolon
\SetKwData{llk}{llk}
\SetKwData{Up}{up}
\SetKwFunction{EmaxApprox}{$\hat E$}
\SetKwFunction{VApprox}{$\hat V$}
\SetKwFunction{LogLik}{$\mathcal{L}$}
\SetKwFunction{CSimulate}{SimulateCostFunction}
\SetKwFunction{InverseGamma}{$InverseGamma$}
\SetKwFunction{Normal}{$N$}
\SetKwFunction{U}{U}
\SetKwFunction{mean}{mean}
\SetKwInOut{Input}{Input}\SetKwInOut{Output}{Output}\SetKwInOut{Variables}{Variables}\SetKwInOut{Initialization}{Initialization}
\Input{$Data$: household characteristics, time of adoption, costs, value of solar. $\beta$ discount factor. $R$ number of MCMC iterations. $\pi_1, \pi_2, \sigma_{\rho}, \sigma_{\alpha}$ and $\sigma_{\gamma}$ hyperparameters to control acceptance rate and convergence of MCMC.}
\Output{$ \left\{\bm\theta^{(r)}\right\}_{r=1}^R = \left\{({\rho}^{(r)}, {\alpha}^{(r)}, {\gamma}^{(r)}, {\sigma}_{\xi}^{(r)})\right\}$ MCMC and posterior mean of structural parameters. For the sake of exposition, assume only one economic and one geographic segment.}
\Variables{$\rho^{(r)}, \alpha^{(r)},\gamma^{(r)}, \xi^{(r)}_i$ accepted values at iteration $r$. $\rho', \alpha',\gamma', \xi'_i$ proposal. $\rho^{0}, \alpha^{0},\gamma^{0}, \xi^{0}_i = 0$}
\For{$r \in 1$ \KwTo $R$}{
\tcp{ ********** HOUSEHOLDS RANDOM EFFECTS **********}
Simulate $c$ and obtain $f$ using Eq \eqref{eq:ct}\;
$\hat E_{it} = \EmaxApprox(\hat V_{it}, f)$, for all $i \in M, t = 1,\ldots, T$, using Eq. \eqref{eq:emax_approx}\;
$l = \LogLik(\rho^{(r-1)}, \alpha^{(r-1)},\gamma^{(r-1)}, \xi^{(r-1)}_i,\hat E_i, Data, \beta, c, f)$, using Eq. \eqref{eq:loglik}\;
$\sigma_{\xi} \sim \InverseGamma(\pi_1, \pi_2)$, using Eq. \eqref{eq:xii}\;
$\xi'_i \sim \Normal(0, \sigma_{\xi})$ using Eq. \eqref{eq:xii}\;
$l'_i = \LogLik(\rho^{(r-1)}, \alpha^{(r-1)},\gamma^{(r-1)}, \xi'_i, \hat E_i, Data, \beta, c, f)$, using Eq. \eqref{eq:loglik}\;
$u \sim \U(0,1)$ \;
\leIf {$\log(u) \leq \min(l'_i - l_i,0)$}{$\xi^{(r)}_i = \xi'_i$}{$\xi^{(r)}_i = \xi^{(r-1)}_i$}
$\hat V_i = \VApprox(\rho^{(r-1)}, \alpha^{(r-1)},\gamma^{(r-1)}, \xi^{(r)}_i,\hat E_i, Data, \beta, c,f)$, using Eq. \eqref{eq:pseudo_vf}\;
\tcp{ ********** BASE UTILITY **********}
Simulate $c$ and obtain $f$ using Eq \eqref{eq:ct}\;
\For{$i \in$ Households}{
$\hat E_i = \EmaxApprox(\hat V_i, f)$, using Eq. \eqref{eq:emax_approx}\;
$l_i = \LogLik(\rho^{(r-1)}, \alpha^{(r-1)},\gamma^{(r-1)}, \xi^{(r)}_i,\hat E_i, Data, \beta, c, f)$ , using Eq. \eqref{eq:loglik}\;
}
$l = \sum l_i$\;
$\rho' \sim \Normal(\rho^{(r-1)}, \sigma_{\rho})$\;
\lFor{$i \in$ Households}{
$l'_i = \LogLik(\rho', \alpha^{(r-1)},\gamma^{(r-1)}, \xi^{(r)}_i,\hat E_i, Data, \beta, c,f)$, using Eq. \eqref{eq:loglik}
}
$l' = \sum l'_i$\;
$u \sim \U(0,1)$ \;
\leIf{$\log(u) \leq \min(l' - l,0)$}{$\rho^{(r)} = \rho'$}{$\rho^{(r)} = \rho^{(r-1)}$}
\lFor{$i \in$ Households}{
$\hat V_i = \VApprox(\rho^{(r)}, \alpha^{(r-1)},\gamma^{(r-1)}, \xi^{(r)}_i,\hat E_i, Data, \beta, c, f)$, using Eq. \eqref{eq:pseudo_vf}\;
}
\tcp{ ********** ECONOMIC FACTOR **********}
Simulate $c$ and obtain $f$ using Eq \eqref{eq:ct}\;
\For{$i \in$ Households}{
$\hat E_i = \EmaxApprox(\hat V_i, f)$, using Eq. \eqref{eq:emax_approx}\;
$l_i = \LogLik(\rho^{(r)}, \alpha^{(r-1)},\gamma^{(r-1)}, \xi^{(r)}_i,\hat E_i, Data, \beta, c, f)$, using Eq. \eqref{eq:loglik}\;
}
$l = \sum l_i$\;
$\alpha' \sim \Normal(\alpha^{(r-1)}, \sigma_{\alpha})$\;
\lFor{$i \in$ Households}{
$l'_i = \LogLik(\rho^{(r)}, \alpha',\gamma^{(r-1)}, \xi^{(r)}_i,\hat E_i, Data, \beta, c, f)$, using Eq. \eqref{eq:loglik}
}
$l' = \sum l'_i$\;
$u \sim \U(0,1)$ \;
\leIf {$\log(u) \leq \min(l' - l,0)$}{$\alpha^{(r)} = \alpha'$}{$\alpha^{(r)} = \alpha^{(r-1)}$}
\lFor{$i \in$ Households}{
$\hat V_i = \VApprox(\rho^{(r)}, \alpha^{(r)},\gamma^{(r-1)}, \xi^{(r)}_i,\hat E_i, Data, \beta, c, f)$, using Eq. \eqref{eq:pseudo_vf}\;
}
\tcp{ ********** NEIGHBORHOOD EFFECTS **********}
Simulate $c$ and obtain $f$ using Eq \eqref{eq:ct}\;
\For{$i \in$ Households}{
$\hat E_i = \EmaxApprox(\hat V_i, f)$, using Eq. \eqref{eq:emax_approx}\;
$l_i = \LogLik(\rho^{(r)}, \alpha^{(r)},\gamma^{(r-1)}, \xi^{(r)}_i,\hat E_i, Data, \beta, c, f)$, using Eq. \eqref{eq:loglik}
}
$l = \sum l_i$\;
$\gamma' \sim \Normal(\gamma^{(r-1)}, \sigma_{\gamma})$\;
\lFor{$i \in$ Households}{
$l'_i = \LogLik(\rho^{(r)}, \alpha^{(r)},\gamma', \xi^{(r)}_i,\hat E_i, Data, \beta, c, f)$, using Eq. \eqref{eq:loglik}
}
$l' = \sum l'_i$\;
$u \sim \U(0,1)$ \;
\leIf {$\log(u) \leq \min(l' - l,0)$}{$\gamma^{(r)} = \gamma'$}{$\gamma^{(r)} = \gamma^{(r-1)}$}
\lFor{$i \in$ Households}{$\hat V_i = \VApprox(\rho^{(r)}, \alpha^{(r)},\gamma^{(r)},\xi^{(r)}_i,\hat E_i, Data, \beta, c, f)$, using Eq. \eqref{eq:pseudo_vf}\;}
}\;
Posterior mean of structural parameters\;
$\bar{\rho} = \mean(\rho^{(r)})$;
$\bar{\alpha} = \mean(\alpha^{(r)})$;
$\bar{\gamma} = \mean(\gamma^{(r)})$;
$\bar{\sigma}_{\xi} = \mean(\sigma^{(r)}_{\xi})$;
\end{algorithm}
}
}
\end{center}
\subsection*{Counterfactual Policy Analysis: Simulation Algorithm}
Simulations based on Bayesian estimation usually propagate every posterior draw through the simulation. However, we use the posterior means $\bar{\bm\theta} = \{\bar{\rho}, \bar{\bm\alpha}, \bar{\bm{\gamma}}, \bar{\bm{\xi}} \}$ as the estimates of the structural parameters to reduce the computational burden because our policy simulation algorithm involves solving for an equilibrium that requires a large number of iterations, as we explain next.
To simulate a specific policy scenario's outcome, we compute the value function and the probabilities of adoption using the fitted value iteration algorithm (see, for example, \cite{Stachurski2009}). Since our best Model (4)H includes only one radial ring, the state variable representing the number of neighboring adopters is a scalar $h_{it}$. Note that, in the dynamic decision problem, $h_{it}$ is endogenous because it is the aggregation of the adoption decisions made by the neighbors of household $i$. Since $h_{it}$ is unobserved in the simulated counterfactual scenarios, we need to compute the dynamic equilibrium to obtain the state transition function for $h_{it}$ based on the household’s rational expectations.
Our simulation algorithm uses an outer loop to compute this state transition equilibrium. We assume that, at each period $t$, household $i$ uses the transition function $h_{i,t+1} = \lambda_i h_{it}$, with $\lambda_i \geq 1$, the conditional mean of the transition in equation \eqref{eq:ht}, to predict the cumulative number of its neighboring adopters in the next period until the total number of adopters equals the number of potential adopters $\hat{h}_i$ in its neighborhood; after reaching this capacity, $h_{it}$ remains constant. Observe that we permit the \textit{growth} rate $\lambda_i$ to vary by household. After simulating the new adopters in each period $t$, we re-calculate $h_{it}$ and re-estimate $\lambda_i$ as the updated state transition function. We iterate this process until the number of adopters per period and $\lambda_{i}$ converge; these iterations constitute the outer loop of our simulation algorithm.
Because the PV system has a finite lifespan of 20 years (80 quarters), inside each outer loop we compute the finite-horizon value function using backward induction. For every household $i$ and $h=1,\ldots,\hat{h}_i$, we first use linear interpolation and the fitted value iteration algorithm to calculate the value function in the last period $t=T$. Then, for any period $t < T$, we use backward induction to calculate the value function. Specifically, the recursive representation of the value function is
\begin{align}
V_{it}(c_t,h_{it},\bm{\varepsilon}_{it}; \bar{\bm\theta})
= \max \left\{\mathcal{V}_{i0t} (c_t,h_{it},\bm{\varepsilon}_{it}; \bar{\bm\theta}),\mathcal{V}_{i1t} (c_t,h_{it},\bm{\varepsilon}_{it}; \bar{\bm\theta})\right\}.
\end{align}
In the simulation, the policy's net-cost schedule $c_t$ is a known input, and household $i$ predicts its next-period neighbor count with the outer-loop transition $\min\{\lambda^{(k)}_i h^{(k)}_{it}, \hat{h}_i\}$, so the continuation value requires no expectation over states. At iteration $k$ of the ENAE algorithm below, the value function is therefore approximated by
\begin{align}
\hat V^{(k)}_{it}({c}_{t},h^{(k)}_{it} ; \bar{\bm\theta}) &
= \log\left(\exp\hat{\mathcal{V}}^{(k)}_{i0t}\left({c}_{t},h^{(k)}_{it}; \bar{\bm\theta}\right)+ \exp\hat{\mathcal{V}}^{(k)}_{i1t}\left({c}_{t},h^{(k)}_{it}; \bar{\bm\theta}\right)\right),
\end{align}
and
\begin{align}
\hat{\mathcal{V}}^{(k)}_{i0t}\left({c}_{t},h^{(k)}_{it} ; \bar{\bm\theta}\right)&
= \beta\, \hat V^{(k)}_{it+1}\left({c}_{t+1}, \min\{\lambda^{(k)}_i h^{(k)}_{it}, \hat{h}_i\}; \bar{\bm\theta}\right)\\
\hat{\mathcal{V}}^{(k)}_{i1t}\left({c}_{t},h^{(k)}_{it} ; \bar{\bm\theta}\right)&
=\bar{\rho} + \bar{\alpha}_{e_i} \textrm{NPV}_{it}\left({c}_t\right) +\bar{\gamma}_{g_i} h^{(k)}_{it} + \bar{\xi}_{i}.
\end{align}
Starting at the last period $T$, we compute each iteration's value function by backward induction. The equilibrium itself is computed by the outer loop: across iterations $k$, the value functions and the implied adopter paths converge to the fixed point of the ENAE algorithm below, $V_i^{(k)} \to V_i$ for each household $i$ and each value of the state $(c_t, h_{it})$. At each iteration, the algorithm records the per-period adoption probabilities $P_{it}$, which embed the probability of not having adopted before $t$, and aggregates them into the expected number of adopters $A^{(k)}_t$.
In our estimation of the model detailed in Section~4.2, we cluster the households with similar characteristics in each census block group, and treat each cluster as a household weighted by the cluster size. Similarly, in the simulation, we compute the adoption probability of the representative household in each cluster, and then calculate the expected number of adopters in the cluster by multiplying this probability by the cluster size.
\textbf{Simulation cost.} A single counterfactual policy path solves the adoption equilibrium below (the fixed point of the ENAE algorithm) in about five outer-loop iterations and roughly 80 minutes of elapsed time on the workstation used for estimation; each policy evaluated in the counterfactual analysis is an independent run of this kind.
\begin{center}
\textbf{Expected Number of Adopters in Equilibrium (ENAE) Algorithm}
\vskip 10pt
{
{\scriptsize
\begin{algorithm}[H]
\SetAlgoLined
\DontPrintSemicolon
\SetKwInOut{Input}{Input}\SetKwInOut{Output}{Output}\SetKwInOut{Variables}{Variables}\SetKwInOut{Initialization}{Initialization}
\Input{Schedule of net cost $c_t, t = 1,\ldots,T$.}
\Output{Expected number of adopters for periods $t = 1,\ldots,T$.}
\Initialization{No household has adopted in period $t = 1$. Iteration $k \leftarrow 1$. $\lambda_i = 1 $ for each household $i$}
\While{the numbers of adopters per period have not converged}{
\For{$i \in$ Households}{
household $i$ adopts in $t = 1$ with probability $P_{i1} = p_{i}(c_1,0;\bar{\bm\theta})$\;
}
\For(\tcp*[f]{the previous loop must finish first}){$i \in$ Households}{
$h^{(k)}_{i1} =$ expected number of adopters at a 1-mile radius of household $i$ at the end of period $t = 1$\;
}
$A^{(k)}_{1} =$ expected number of adopters in period $1$ of iteration $k$\;
\For{$t \in 2$ \KwTo $T$}{
\For{$i \in$ Households that have not yet adopted}{
household $i$ adopts in period $t$ with probability $P_{it} = p_{i}(c_t,h^{(k)}_{it-1};\bar{\bm\theta}) * (1 - \sum_{\tau = 1}^{t-1}P_{i\tau})$\;
}
\For{$i \in$ Households}{
$h^{(k)}_{it} =$ expected number of adopters at a 1-mile radius of household $i$ at the end of period $t$\;
}
$A^{(k)}_{t} =$ expected number of adopters in period $t$ of iteration $k$\;
}
\For{$i \in$ Households}{
Estimate linear regression to update $\lambda_i$: $h^{(k)}_{it+1} = \lambda_i h^{(k)}_{it}$\;
}
$k \leftarrow k+1$;
}
\end{algorithm}
}
}
\end{center}
\clearpage
\section{Supporting Figures for Bayesian Posterior Sampling and Policy Simulations} \label{sec:SupportingFigures}
\textbf{Retained sample and convergence diagnostics.} We run the sampler in Appendix C for 18{,}000 iterations under a 4-hour wall-clock limit on a single processor. The chain reaches its stationary distribution within the first 2{,}500 iterations, which we discard as burn-in, and we compute the posterior means and 95\% credible intervals of the structural parameters from the retained draws; the parameter trace and density plots below display the chains and confirm convergence. Mixing differs across parameters: the heterogeneity scale and the majority-segment net-present-value coefficient yield effective sample sizes in the thousands, whereas the remaining coefficients, and the neighborhood-effect coefficients in particular, are more strongly autocorrelated and yield effective sample sizes in the tens to low hundreds; we therefore run each chain well past burn-in rather than stopping once stationarity is reached. The reported estimates are stable under longer sampling: extending the run from 4 to 8 hours leaves each posterior mean within roughly a quarter of a posterior standard deviation of its reported value, with the exception of the weakly-mixed urban neighborhood-effect coefficient, whose mean shifts by about a third of a standard deviation, consistent with its low effective sample size.
\begin{figure}[H]
{ \centering
\includegraphics[scale=0.42]{plot_trace_plots.pdf}
\caption{MCMC trace plots of the best model, Model (4)H, structural parameters.\label{fig:mcmc1_exVIa-RE}}
}
{\footnotesize \setlength{\baselineskip}{10pt}
\noindent \emph{Note.} From top to bottom: (a) $\rho$, the constant term of the utility function; (b) $\alpha$, the preference for the economic benefit of solar per economic segment; (c) $\gamma$, the neighborhood effects per geographic segment; and (d) $\sigma_{\xi}$, the standard deviation of the household random effects. \textit{Note}: the first 2{,}500 iterations, the \textit{burn-in} period, are discarded because the initial samples may not accurately represent the target distribution due to the starting point of the chain.
\par}
\end{figure}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.95\textwidth]{RSP40_trajectory}
\caption{Counterfactual trajectory under the one-step rebate switching policy RSP(40).\label{fig:ECrsp40}}
\end{center}
{\footnotesize \setlength{\baselineskip}{10pt}
\noindent \emph{Note.} (a) net cost per watt; (b) cumulative adopters. The adoption rate rises as the pre-announced rebate switching period approaches and drops once the rebate ends, producing the kink at period~40 discussed in Section~6.1 of the paper.
\par}
\end{figure}
\clearpage
\section{Additional Model Comparison and Selection Results}\label{sec:DetailResultsTables}
In this appendix, we report the results from the additional models we tested. To assess the robustness of our model selection, we compare out-of-sample predictive performance using both the mean absolute error (MAE) and the mean absolute percentage error (MAPE). The results are reported in Table~\ref{tab:summaryResultsAllModels}.
\begin{table}[H]
\begin{center}
\caption{Comparison of various model specifications.\label{tab:summaryResultsAllModels}}
{\scriptsize
\begin{tabular}{lllrrrr}
\toprule
& \multicolumn{1}{c}{\textbf{Economic Segment (${E}$)}} & \multicolumn{1}{c}{\textbf{Neighborhood Radial Ring (${D}$)}} & \multicolumn{2}{c} {\textbf{MAE (StdMAE)}} & \multicolumn{2}{c} {\textbf{MAPE (StdMAPE) [\%]}}\bigstrut\\
\cline{4-5} \cline{6-7}
\multicolumn{1}{c}{Model} & \multicolumn{1}{p{3cm}}{\centering Home Mkt Value (\$’000) } & \multicolumn{1}{p{3cm}}{\centering Radial Distance Brackets (Miles)} & \multicolumn{1}{c}{NH} & \multicolumn{1}{c}{H}& \multicolumn{1}{c}{NH} & \multicolumn{1}{c}{H} \bigstrut\\\midrule
(1) & [0,$\infty$) & no neighborhood effects &
67.726 (5.994) & 71.482 (4.974) &
29.778 (2.852) & 31.552 (2.373)
\bigstrut\\
(2) & [0,$\infty$) & [0,1) &
37.343 (12.247) & 32.656 (6.721) &
19.314 (6.036) & 16.956 (3.496)
\bigstrut \\
(3) & [0,300),[300,$\infty$) & no neighborhood effects &
85.001 (3.871) & 85.511 (3.661) &
38.144 (1.861) & 38.386 (1.764)
\bigstrut \\
(4) & [0,300),[300,$\infty$) & [0,1) &
45.768 (14.155) & 31.845 (2.412) &
23.407 (6.641) & 15.599 (1.552)
\bigstrut \\
(5) & [0,300),[300,600),[600,$\infty$) & no neighborhood effects &
69.011 (4.542) & 69.627 (4.353) &
30.889 (2.151) & 31.177 (2.061)
\bigstrut\\
(6) & [0,300),[300,600),[600,$\infty$) & [0,1) &
48.534 (5.587) & 49.142 (5.437) &
23.904 (2.925) & 24.118 (2.797)
\bigstrut\\\hline
(7) & [0,$\infty$) & [0,0.5) &
37.331 (4.904) & 32.106 (2.006) &
16.591 (1.703) & 15.074 (1.161)
\bigstrut \\
(8) & [0,$\infty$) & [0,0.5),[0.5,1) &
33.516 (6.168) & 30.550 (4.012) &
16.709 (3.001) & 15.925 (2.168)
\bigstrut \\
(9) & [0,$\infty$) & [0,1),[1,SB) &
37.969 (6.740) & 38.463 (6.621) &
16.375 (2.936) & 16.503 (2.942)
\bigstrut \\
(10) & [0,300),[300,$\infty$) & [0,0.5) &
31.651 (1.597) & 31.373 (1.180) &
14.479 (0.931) & 14.197 (0.638)
\bigstrut \\
(11) & [0,300),[300,$\infty$) & [0,0.5),[0.5,1) &
33.055 (3.929) & 31.826 (2.613)&
17.141 (2.244) & 16.028 (1.534)
\bigstrut \\
(12) & [0,300),[300,$\infty$) & [0,1),[1,SB) &
38.152 (6.754) & 39.365 (7.304)&
16.576 (3.046) & 17.112 (3.319)
\bigstrut \\
(13) & [0,300),[300,600),[600,$\infty$) & [0,0.5) &
44.348 (3.174) & 43.184 (3.069) &
20.972 (1.779) & 20.386 (1.741)
\bigstrut \\
(14) & [0,300),[300,600),[600,$\infty$) & [0,0.5),[0.5,1) &
51.830 (7.820) & 48.625 (6.994)&
26.044 (3.714) & 24.492 (3.421)
\bigstrut \\
(15) & [0,300),[300,600),[600,$\infty$) & [0,1),[1,SB) &
37.077 (3.032) & 36.472 (2.768) &
16.473 (1.699) & 16.166 (1.577)
\bigstrut[b] \\
\bottomrule
\multicolumn{7}{l}{Note: (a) NH: without heterogeneity, H: with heterogeneity;}\\
\multicolumn{7}{l}{$\quad\quad~~$(b) Training periods: 1-33, test periods: 34-38;}\\
\multicolumn{7}{l}{$\quad\quad~~$(c) In estimation: $\hat{h}_{it+1} = h_{it+1}$, in prediction: $\hat{h}_{it+1} = \lambda_i \tilde{h}_{it}$;}\\
\multicolumn{7}{l}{$\quad\quad~~$(d) MAE is calculated from 1000 simulations for out-of-sample periods set $\mathcal{T}$ as the average absolute error between} \\
\multicolumn{7}{l}{$\quad\quad\quad~~~~$the actual number of adopters $N_t$ in period $t$ and the predicted number $\hat{N}^k_t$ in simulation $k$ and period $t$,} \\
\multicolumn{7}{l}{$\quad\quad\quad~~~~$measured in number of adopters; MAPE is the corresponding mean absolute percentage error.
} \\
\end{tabular}\\
}
\end{center}
\end{table}
\clearpage
\subsection*{Stability of the structural parameters across specifications}\label{sec:ECparamstability}
This section tests the robustness of our results to the model specification, extending the comparison in Table~\ref{tab:summaryResultsAllModels} across the economic segmentations $E$ and the neighborhood radial rings $D$. Two facts anchor the comparison. First, extending the neighborhood beyond one mile adds only a negligible neighborhood effect: in the two-ring models (9), (12), and (15), the coefficient on adopters beyond one mile is near $-4\times10^{-4}$, three orders of magnitude below the inner-ring coefficient (Table~\ref{tab:ECparamGamma}). Second, the leading one-mile and half-mile specifications predict almost equally well: their out-of-sample MAEs differ by less than 2\% (for the two-segment models, 31.85 for Model~(4)H against 31.37 for Model~(10)H). Since predictive accuracy alone does not separate these specifications, we examine the structural parameter estimates, reported for the twelve heterogeneity (H) specifications with neighborhood effects, models (2), (4), (6), and (7)--(15).
The estimates are stable across these alternative specifications (Tables~\ref{tab:ECparamRhoAlpha} and~\ref{tab:ECparamGamma} and Figure~\ref{fig:ECparamstability}). The heterogeneity standard deviation $\sigma_{\xi}$ is statistically indistinguishable across the twelve specifications, its $95\%$ credible intervals all overlapping near $0.051$. The baseline utility $\rho$ stays within the narrow band $[-6.85,-6.64]$ and is precisely estimated (low posterior variance) in every specification. The net-present-value coefficient $\alpha$ is larger for higher-value home segments wherever the segment boundaries align, so higher-value households ascribe more weight to the economic benefit of a PV system: the low-value coefficient $\alpha_{[0,300)}$ has eight mutually overlapping $95\%$ credible intervals around $3\times10^{-4}$, the two-segment high-value coefficient overlapping intervals around $6.2\times10^{-4}$, and the three-segment $[300,600)$ coefficient overlapping intervals around $8.5\times10^{-4}$, so no specification's estimate is distinguishable from the others at the $95\%$ level; the single-segment models, which pool all home values into one economic segment, estimate a single coefficient near $4.1\times10^{-4}$. The only imprecisely identified coefficient is the top $[600,\infty)$ bin, whose credible intervals are wide and rest on few very-high-value adopters, consistent with the expanding-window evidence in Appendix~F. The neighborhood effect preserves the ordering $\gamma_{\mathrm{rural}}>\gamma_{\mathrm{suburban}}>\gamma_{\mathrm{urban}}$ in every one-mile-ring specification, and a ring standing alone recovers the same one-mile coefficients as when it is the inner ring of a two-ring model, the standalone and inner-ring $\gamma_{\mathrm{rural}}$ intervals overlapping (means $0.121$ and $0.135$).
The neighborhood coefficient does scale with the ring width, which is mechanical: a half-mile ring aggregates far fewer installed systems, so the per-neighbor coefficient is larger and, in sparsely built rural areas, noisier (the half-mile rural coefficient carries wider credible intervals, for example $(0.13,0.40)$ in Model~(8) against $(0.11,0.16)$ for the one-mile ring in Model~(4)). This is why the half-mile specifications that match Model~(4)H on out-of-sample error, such as Model~(10), do so with a less stable and less interpretable near-neighbor coefficient and a geographic ordering that is no longer monotone. The one-mile ring therefore delivers the same forecast accuracy to within the noise while giving a well-identified, monotone neighborhood structure, and we use the one-mile radial ring, with two economic segments (Model~(4)H), as the parsimonious specification of our model.
\begin{table}[H]
\caption{Posterior mean and 95\% credible interval of $\rho, \sigma_{\xi},\alpha$ of Models (2), (4), and (6) to (15).\label{tab:ECparamRhoAlpha}}
{\centering
\footnotesize
\setlength{\tabcolsep}{4pt} \resizebox{\textwidth}{!}{
\begin{tabular}{clccccc}
\toprule
Model & Economic segments $E$ & $\rho$ & $\sigma_{\xi}$ & $\alpha_1$ & $\alpha_2$ & $\alpha_3$ \\
\midrule
(2) & $[0,\infty)$ & $-$6.80990 & 0.05094 & 0.00041 & & \\
& & {\scriptsize ($-$6.94269, $-$6.70937)} & {\scriptsize (0.04839, 0.05364)} & {\scriptsize (0.00038, 0.00047)} & & \\
\addlinespace[3pt]
(4) & $[0,300),[300,\infty)$ & $-$6.79158 & 0.05098 & 0.00029 & 0.00064 & \\
& & {\scriptsize ($-$6.89602, $-$6.71202)} & {\scriptsize (0.04851, 0.05368)} & {\scriptsize (0.00026, 0.00032)} & {\scriptsize (0.00059, 0.00070)} & \\
\addlinespace[3pt]
(6) & $[0,300),[300,600),[600,\infty)$ & $-$6.81902 & 0.05180 & 0.00029 & 0.00086 & 0.00052 \\
& & {\scriptsize ($-$6.91498, $-$6.73623)} & {\scriptsize (0.04924, 0.05443)} & {\scriptsize (0.00026, 0.00033)} & {\scriptsize (0.00081, 0.00093)} & {\scriptsize (0.00024, 0.00097)} \\
\addlinespace[3pt]
(7) & $[0,\infty)$ & $-$6.74818 & 0.05100 & 0.00043 & & \\
& & {\scriptsize ($-$6.83022, $-$6.67359)} & {\scriptsize (0.04852, 0.05378)} & {\scriptsize (0.00039, 0.00049)} & & \\
\addlinespace[3pt]
(8) & $[0,\infty)$ & $-$6.81625 & 0.05102 & 0.00041 & & \\
& & {\scriptsize ($-$6.90705, $-$6.72905)} & {\scriptsize (0.04848, 0.05360)} & {\scriptsize (0.00037, 0.00045)} & & \\
\addlinespace[3pt]
(9) & $[0,\infty)$ & $-$6.63613 & 0.05189 & 0.00041 & & \\
& & {\scriptsize ($-$6.72372, $-$6.54712)} & {\scriptsize (0.04929, 0.05467)} & {\scriptsize (0.00038, 0.00045)} & & \\
\addlinespace[3pt]
(10) & $[0,300),[300,\infty)$ & $-$6.74915 & 0.05109 & 0.00031 & 0.00062 & \\
& & {\scriptsize ($-$6.82449, $-$6.67770)} & {\scriptsize (0.04850, 0.05379)} & {\scriptsize (0.00028, 0.00034)} & {\scriptsize (0.00058, 0.00068)} & \\
\addlinespace[3pt]
(11) & $[0,300),[300,\infty)$ & $-$6.81931 & 0.05107 & 0.00030 & 0.00062 & \\
& & {\scriptsize ($-$6.90434, $-$6.74042)} & {\scriptsize (0.04863, 0.05375)} & {\scriptsize (0.00026, 0.00033)} & {\scriptsize (0.00058, 0.00068)} & \\
\addlinespace[3pt]
(12) & $[0,300),[300,\infty)$ & $-$6.64954 & 0.05114 & 0.00029 & 0.00060 & \\
& & {\scriptsize ($-$6.73396, $-$6.56284)} & {\scriptsize (0.04864, 0.05385)} & {\scriptsize (0.00026, 0.00032)} & {\scriptsize (0.00057, 0.00065)} & \\
\addlinespace[3pt]
(13) & $[0,300),[300,600),[600,\infty)$ & $-$6.78787 & 0.05099 & 0.00032 & 0.00086 & 0.00034 \\
& & {\scriptsize ($-$6.86890, $-$6.71528)} & {\scriptsize (0.04842, 0.05369)} & {\scriptsize (0.00029, 0.00035)} & {\scriptsize (0.00081, 0.00093)} & {\scriptsize (0.00017, 0.00074)} \\
\addlinespace[3pt]
(14) & $[0,300),[300,600),[600,\infty)$ & $-$6.84883 & 0.05097 & 0.00030 & 0.00085 & 0.00026 \\
& & {\scriptsize ($-$6.93669, $-$6.76698)} & {\scriptsize (0.04848, 0.05362)} & {\scriptsize (0.00027, 0.00034)} & {\scriptsize (0.00080, 0.00091)} & {\scriptsize (0.00015, 0.00041)} \\
\addlinespace[3pt]
(15) & $[0,300),[300,600),[600,\infty)$ & $-$6.68338 & 0.05106 & 0.00029 & 0.00082 & 0.00033 \\
& & {\scriptsize ($-$6.76686, $-$6.59498)} & {\scriptsize (0.04857, 0.05379)} & {\scriptsize (0.00026, 0.00033)} & {\scriptsize (0.00078, 0.00087)} & {\scriptsize (0.00016, 0.00061)} \\
\bottomrule
\end{tabular}}}
\vspace{0.2cm}
{\footnotesize
\setlength{\baselineskip}{10pt}
\noindent \emph{Note.}
{Posterior mean and 95\% credible interval of the baseline utility $\rho$, the heterogeneity standard deviation $\sigma_{\xi}$, and the net-present-value coefficient $\alpha$ across specifications. The coefficients $\alpha_1,\alpha_2,\alpha_3$ correspond, in order, to the economic segments listed in $E$.}
{Empty cells indicate segments not present in that model.}
\par}
\end{table}
\begin{table}[H]
\caption{Posterior mean and 95\% credible interval of $\gamma$ of Models (2), (4), and (6) to (15).\label{tab:ECparamGamma}}
{\centering\footnotesize
\setlength{\tabcolsep}{4pt} \resizebox{\textwidth}{!}{
\begin{tabular}{clcccccc}
\toprule
& & \multicolumn{3}{c}{Inner ring} & \multicolumn{3}{c}{Outer ring}\\
\cmidrule(lr){3-5}\cmidrule(lr){6-8}
Model & Rings $D$ & urban & suburban & rural & urban & suburban & rural \\
\midrule
(2) & $[0,1)$ & 0.04703 & 0.05135 & 0.13372 & & & \\
& & {\scriptsize (0.02800, 0.08690)} & {\scriptsize (0.04685, 0.05984)} & {\scriptsize (0.11403, 0.19105)} & & & \\
\addlinespace[3pt]
(4) & $[0,1)$ & 0.03611 & 0.04932 & 0.12135 & & & \\
& & {\scriptsize (0.02695, 0.05801)} & {\scriptsize (0.04518, 0.05853)} & {\scriptsize (0.10514, 0.15563)} & & & \\
\addlinespace[3pt]
(6) & $[0,1)$ & 0.03726 & 0.04689 & 0.10768 & & & \\
& & {\scriptsize (0.02696, 0.06488)} & {\scriptsize (0.04338, 0.05249)} & {\scriptsize (0.09368, 0.13662)} & & & \\
\addlinespace[3pt]
(7) & $[0,0.5)$ & 0.14120 & 0.06114 & 0.35359 & & & \\
& & {\scriptsize (0.12194, 0.17290)} & {\scriptsize (0.05654, 0.06987)} & {\scriptsize (0.30477, 0.46793)} & & & \\
\addlinespace[3pt]
(8) & $[0,0.5),[0.5,1)$ & 0.12687 & 0.05445 & 0.21106 & 0.01648 & 0.04654 & 0.10549 \\
& & {\scriptsize (0.09766, 0.16437)} & {\scriptsize (0.04946, 0.06066)} & {\scriptsize (0.12858, 0.40122)} & {\scriptsize (0.00339, 0.03799)} & {\scriptsize (0.03185, 0.07586)} & {\scriptsize (0.05398, 0.16656)} \\
\addlinespace[3pt]
(9) & $[0,1),[1,\infty)$ & 0.03440 & 0.04932 & 0.13863 & $-$0.00037 & $-$0.00015 & $-$0.00035 \\
& & {\scriptsize (0.02905, 0.04308)} & {\scriptsize (0.04576, 0.05456)} & {\scriptsize (0.12058, 0.16632)} & {\scriptsize ($-$0.00057, $-$0.00019)} & {\scriptsize ($-$0.00029, $-$0.00001)} & {\scriptsize ($-$0.00060, $-$0.00009)} \\
\addlinespace[3pt]
(10) & $[0,0.5)$ & 0.13585 & 0.05878 & 0.32864 & & & \\
& & {\scriptsize (0.11765, 0.17079)} & {\scriptsize (0.05438, 0.06551)} & {\scriptsize (0.28539, 0.40965)} & & & \\
\addlinespace[3pt]
(11) & $[0,0.5),[0.5,1)$ & 0.11830 & 0.05309 & 0.19035 & 0.01387 & 0.04687 & 0.09366 \\
& & {\scriptsize (0.09466, 0.14641)} & {\scriptsize (0.04874, 0.05987)} & {\scriptsize (0.11599, 0.27162)} & {\scriptsize (0.00445, 0.03165)} & {\scriptsize (0.03127, 0.07442)} & {\scriptsize (0.05600, 0.14257)} \\
\addlinespace[3pt]
(12) & $[0,1),[1,\infty)$ & 0.03578 & 0.04764 & 0.13672 & $-$0.00039 & $-$0.00013 & $-$0.00046 \\
& & {\scriptsize (0.02994, 0.04855)} & {\scriptsize (0.04436, 0.05201)} & {\scriptsize (0.11822, 0.16704)} & {\scriptsize ($-$0.00067, $-$0.00020)} & {\scriptsize ($-$0.00026, 0.00001)} & {\scriptsize ($-$0.00074, $-$0.00016)} \\
\addlinespace[3pt]
(13) & $[0,0.5)$ & 0.13146 & 0.05690 & 0.30728 & & & \\
& & {\scriptsize (0.11391, 0.16309)} & {\scriptsize (0.05219, 0.06477)} & {\scriptsize (0.26092, 0.42666)} & & & \\
\addlinespace[3pt]
(14) & $[0,0.5),[0.5,1)$ & 0.11554 & 0.05017 & 0.17273 & 0.01577 & 0.04415 & 0.08735 \\
& & {\scriptsize (0.09113, 0.14541)} & {\scriptsize (0.04676, 0.05502)} & {\scriptsize (0.08742, 0.26271)} & {\scriptsize (0.00499, 0.03407)} & {\scriptsize (0.03084, 0.07583)} & {\scriptsize (0.04775, 0.14083)} \\
\addlinespace[3pt]
(15) & $[0,1),[1,\infty)$ & 0.03388 & 0.04539 & 0.13092 & $-$0.00037 & $-$0.00011 & $-$0.00052 \\
& & {\scriptsize (0.02893, 0.04551)} & {\scriptsize (0.04277, 0.04905)} & {\scriptsize (0.11465, 0.15367)} & {\scriptsize ($-$0.00061, $-$0.00017)} & {\scriptsize ($-$0.00023, 0.00002)} & {\scriptsize ($-$0.00078, $-$0.00025)} \\
\bottomrule
\end{tabular}}}
\vspace{0.2cm}
{\footnotesize
\setlength{\baselineskip}{10pt}
\noindent \emph{Note.} Posterior mean and 95\% credible interval of the neighborhood effect $\gamma$ by geographic segment and radial ring across specifications. Inner and outer denote the first and second rings in $D$; single-ring models have no outer ring, and the beyond-one-mile (outer) effects in models (9), (12), and (15) are negligible, three orders of magnitude below the inner-ring effects.
\par}
\end{table}
\begin{figure}[H]
{ \centering
\includegraphics[width=\textwidth]{param_stability.pdf}
\caption{Stability of the structural parameters across economic-segment and radial-ring specifications. \label{fig:ECparamstability}}
}
{\footnotesize
\setlength{\baselineskip}{9pt}
(a) baseline utility $\rho$, with the dashed line the mean and the shaded band the range; (b) the net-present-value coefficient $\alpha$ by economic segment; (c) the neighborhood effect $\gamma$ for the inner ring by geographic segment. Whiskers are $95\%$ credible intervals. Models left of the dotted line use a one-mile inner ring; those to the right use a half-mile inner ring.
\noindent \emph{Note.}
\par}
\end{figure}
\begin{table}[H]
\begin{center}
\caption{Predicted performance of models using alternative function specifications of $\textbf{h}$ in the utility function.\label{tab:summaryResults3}}
{\scriptsize
\begin{tabular}{ccc}
\toprule
\multicolumn{1}{c}{\textbf{Specification of $\textbf{h}$ in utility}} & \multicolumn{1}{c} {\textbf{MAE (StdMAE)}} & \multicolumn{1}{c} {\textbf{MAPE (StdMAPE) [\%]}}\bigstrut\\
\midrule
$h$ & 31.845 (2.412)& 15.599 (1.552)
\bigstrut\\
$\log(1+h)$ & 106.901 (6.648) & 54.746 (4.221) \bigstrut \\
$\sqrt{h}$ & 74.434 (5.631) & 37.672 (4.208) \bigstrut \\
\bottomrule
\multicolumn{3}{l}{Note: (a) Training periods: 1-33, test periods: 34-38;}\\
\multicolumn{3}{l}{$\quad\quad~~$(b) In estimation: $\hat{h}_{it+1} = h_{it+1}$, in prediction: $\hat{h}_{it+1} = \lambda_i \hat{h}_{it}$;}\\
\multicolumn{3}{l}{$\quad\quad~~$(c) Economic Segments (${E}$): $[0,300),[300, \infty)$;}\\
\multicolumn{3}{l}{$\quad\quad~~$(d) Neighborhood Radial Ring (${D}$): $[0,1)$.}\\
\end{tabular}
}
\end{center}
\end{table}
\clearpage
\section{Estimation with Expanding Data Window}\label{sec:ECexpandingwindow}
We assess the consistency of the estimates through an expanding-window re-estimation. We re-estimate the selected model, Model~(4)H, on the first $T$ quarters of the sample at even windows from $T=26$ (2010Q2) through $T=36$ (2012Q4), together with the full estimation window $T=33$ (2012Q1) reported in the paper, and track the posterior mean and 95\% credible interval of each structural parameter as the window grows. Figure~\ref{fig:ECexpwindow} plots the trajectories grouped by parameter family, on axes wide enough to span the estimates at every window; Table~\ref{tab:ECexpwindow} reports the underlying per-window estimates.
The net-present-value coefficients $\alpha$ of the two economic segments cross between windows $T=30$ and $T=32$ (Figure~\ref{fig:ECexpwindow}b). In the early windows ($T\le 30$, through 2011Q2) the coefficient for the high-value $[300\mathrm{K},\infty)$ segment sits below that of the majority $[0,300\mathrm{K})$ segment, estimated near $2\times10^{-4}$: few high-value households have adopted by then, so their net-present-value coefficient has greater estimation error, reflected in wider interval estimates. As the window extends through the high-adoption quarters of 2011--2012, more high-value adoptions enter the sample and $\alpha_{[300\mathrm{K},\infty)}$ rises, crossing above $\alpha_{[0,300\mathrm{K})}$ at $T=32$ and settling near its full-sample value of $6.4\times10^{-4}$ at $T=33$. The crossover marks improved identification of the high-value segment's price response rather than a change in preferences: once enough high-value adoptions are observed, the ordering $\alpha_{[300\mathrm{K},\infty)}>\alpha_{[0,300\mathrm{K})}$ reported for the full sample is recovered and holds for every window from $T=32$ onward.
The remaining parameters support the stability reported in Section~5 of the paper (Figure~\ref{fig:ECexpwindow}; Table~\ref{tab:ECexpwindow}). The standard deviation of household heterogeneity $\sigma_{\xi}$ varies only between $0.0504$ and $0.0515$, the baseline utility $\rho$ stays within a narrow band ($-7.19$ to $-6.76$), and the coefficient of the majority segment $\alpha_{[0,300\mathrm{K})}$ holds its order of magnitude, near $3$ to $4\times10^{-4}$. The coefficients $\gamma$ for neighborhood effects are a little more sensitive to the sample sizes: their estimates are larger in the short early windows, with wider interval estimates, and gradually level off as the larger sampling window absorbs more data, when falling net costs account for a growing share of adoption. Their economic interpretation is nevertheless invariant over time: the ordering $\gamma_{\mathrm{rural}}>\gamma_{\mathrm{suburban}}>\gamma_{\mathrm{urban}}$ is preserved and no coefficient changes sign. By the full estimation window $T=33$, the heterogeneity scale, the baseline utility, and the majority-segment coefficient have settled and the high-value coefficient is identified, the stability a policymaker needs before relying on the estimates for counterfactual analysis.
\begin{table}[!h]
\caption{Posterior mean and 95\% credible interval of the structural parameters of Model~(4)H across expanding estimation windows\label{tab:ECexpwindow}}
{\centering
{\footnotesize
\setlength{\tabcolsep}{4pt}\resizebox{\textwidth}{!}{
\begin{tabular}{lccccccc}
\toprule
Parameter & $T=26$ & $T=28$ & $T=30$ & $T=32$ & \textbf{$T=33$} & $T=34$ & $T=36$ \\
\midrule
$\rho$ & $-$7.10438 & $-$7.14538 & $-$7.19179 & $-$7.04018 & \textbf{$-$6.79158} & $-$6.76025 & $-$6.84665 \\
& {\scriptsize ($-$7.19273, $-$7.01396)} & {\scriptsize ($-$7.22766, $-$7.07685)} & {\scriptsize ($-$7.26856, $-$7.10772)} & {\scriptsize ($-$7.11829, $-$6.96704)} & {\scriptsize ($-$6.89602, $-$6.71202)} & {\scriptsize ($-$6.82481, $-$6.69228)} & {\scriptsize ($-$6.90845, $-$6.78334)} \\
\addlinespace[3pt]
$\alpha_{[0,300\mathrm{K})}$ & 0.00040 & 0.00039 & 0.00042 & 0.00033 & \textbf{0.00029} & 0.00028 & 0.00040 \\
& {\scriptsize (0.00036, 0.00044)} & {\scriptsize (0.00036, 0.00043)} & {\scriptsize (0.00038, 0.00046)} & {\scriptsize (0.00030, 0.00037)} & {\scriptsize (0.00026, 0.00032)} & {\scriptsize (0.00025, 0.00031)} & {\scriptsize (0.00037, 0.00043)} \\
\addlinespace[3pt]
$\alpha_{[300\mathrm{K},\infty)}$ & 0.00023 & 0.00018 & 0.00025 & 0.00065 & \textbf{0.00064} & 0.00076 & 0.00088 \\
& {\scriptsize (0.00017, 0.00031)} & {\scriptsize (0.00012, 0.00027)} & {\scriptsize (0.00019, 0.00036)} & {\scriptsize (0.00060, 0.00070)} & {\scriptsize (0.00059, 0.00070)} & {\scriptsize (0.00071, 0.00082)} & {\scriptsize (0.00085, 0.00090)} \\
\addlinespace[3pt]
$\gamma_{\mathrm{urban}}$ & 0.07531 & 0.05378 & 0.05164 & 0.04569 & \textbf{0.03611} & 0.02430 & 0.01330 \\
& {\scriptsize (0.06744, 0.08689)} & {\scriptsize (0.04792, 0.06109)} & {\scriptsize (0.04627, 0.05929)} & {\scriptsize (0.03765, 0.06005)} & {\scriptsize (0.02695, 0.05801)} & {\scriptsize (0.01856, 0.03579)} & {\scriptsize (0.00901, 0.02063)} \\
\addlinespace[3pt]
$\gamma_{\mathrm{suburban}}$ & 0.11858 & 0.08577 & 0.11679 & 0.07235 & \textbf{0.04932} & 0.03979 & 0.02209 \\
& {\scriptsize (0.10389, 0.14250)} & {\scriptsize (0.07663, 0.09954)} & {\scriptsize (0.10990, 0.12572)} & {\scriptsize (0.06793, 0.07957)} & {\scriptsize (0.04518, 0.05853)} & {\scriptsize (0.03732, 0.04318)} & {\scriptsize (0.02000, 0.02493)} \\
\addlinespace[3pt]
$\gamma_{\mathrm{rural}}$ & 0.21835 & 0.15478 & 0.15084 & 0.13673 & \textbf{0.12135} & 0.17655 & 0.07860 \\
& {\scriptsize (0.19393, 0.25007)} & {\scriptsize (0.14232, 0.17272)} & {\scriptsize (0.13646, 0.17118)} & {\scriptsize (0.12302, 0.15766)} & {\scriptsize (0.10514, 0.15563)} & {\scriptsize (0.16047, 0.20891)} & {\scriptsize (0.07068, 0.09236)} \\
\addlinespace[3pt]
$\sigma_{\xi}$ & 0.05051 & 0.05039 & 0.05052 & 0.05079 & \textbf{0.05098} & 0.05126 & 0.05153 \\
& {\scriptsize (0.04802, 0.05313)} & {\scriptsize (0.04806, 0.05283)} & {\scriptsize (0.04801, 0.05320)} & {\scriptsize (0.04824, 0.05364)} & {\scriptsize (0.04851, 0.05368)} & {\scriptsize (0.04856, 0.05392)} & {\scriptsize (0.04909, 0.05421)} \\
\addlinespace[3pt]
\bottomrule
\end{tabular}}}}
\vspace{0.2cm}
{\footnotesize
\setlength{\baselineskip}{10pt}
\noindent \emph{Note.}
Column $T$ re-estimates the model on quarters $1$ to $T$; the $T=33$ column (boldface) reproduces the full-sample estimates reported in Table~5 of the paper.
$\rho$ baseline utility; $\alpha$ net-present-value coefficient by home-value segment ($[0,300\mathrm{K})$ and $[300\mathrm{K},\infty)$); $\gamma$ neighborhood effect for a one-mile ring by geographic segment(urban, suburban, rural); $\sigma_{\xi}$ standard deviation of household heterogeneity. Calendar mapping: $T=26$ is 2010Q2, $T=33$ is 2012Q1, and $T=36$ is 2012Q4.
\par}
\end{table}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.8\textwidth]{ExpandingWindow_ParamTrajectories_2x2_26to36.pdf}
\caption{Expanding-window estimates of the structural parameters of Model~(4)H. \label{fig:ECexpwindow}}
\end{center}
{\footnotesize
\setlength{\baselineskip}{10pt}
\noindent \emph{Note.}
Expanding-window estimates of the structural parameters of Model~(4)H over even windows $T=26$ to $36$ together with the full estimation window $T=33$ reported in the paper. Each panel shows the posterior mean (markers) and 95\% credible interval (shaded band).
\par}
\end{figure}
\clearpage
\section{Benchmark Model Details}\label{sec:BassModel}
\subsection*{Benchmark (a): Hazard model (individual level)}
This is the reduced-form hazard model applied in \citet{Bollinger2022visibility}, estimated on our data. The discrete hazard benchmark model (15) in Section 5.1 is estimated
by maximum likelihood estimation (MLE), treating PV adoption as $\{0,1\}$ dummy variables over time
in a discrete-time hazard process. For each household-period observation
in which household $i$ has not yet adopted, define $y_{it}=1$
if the household adopts in period $t$, and $y_{it}=0$ otherwise.
The model specifies the conditional adoption probability, or hazard
rate, as
\[
G\left(\lambda_{it}\right)=\xi_{z}+\eta_{t}+\beta_{1}\text{NPV}_{it}+\beta_{2}X_{i}+\beta_{3}H_{it}+\beta_{4}\text{NPV}_{H_{it}},
\]
\noindent where $G(\cdot)$ is either the logit link $\log\!\left(\lambda_{it}/(1-\lambda_{it})\right)$
or the complementary log-log link $\log\!\left(-\log(1-\lambda_{it})\right)$, both standard
in discrete-time hazard models \citep{Kalbfleisch2002a}; $\xi_{z}$ are zip-code fixed
effects that capture market and geographic factors, $\eta_{t}$ is the temporal effect,
modeled flexibly (polynomial, B-spline, natural spline, or local regression), and $X_{i}$
collects the household characteristics: the total footprint of the house, the ratio of tree
cover to footprint, the sunlight received, the hillshade, and the solar irradiance. The
variables $H_{it}$ and $\text{NPV}_{H_{it}}$ denote, respectively, the cumulative number of peer
installations and the cumulative economic value (NPV) of those installations within the
one-mile radial ring. The likelihood contribution for household $i$ in period $t$ is
therefore $\lambda_{it}^{y_{it}}(1-\lambda_{it})^{1-y_{it}}$.
The MLE maximizes the log-likelihood
\[
\sum_{i,t}\left(y_{it}\log(\lambda_{it})+(1-y_{it})\log(1-\lambda_{it})\right),
\]
over all household-period observations in the risk set by choosing
the optimal coefficients $\left\{ \beta_{1},\ldots,\beta_{4}\right\}$ and
fixed effects $\left\{ \xi_{z},\eta_{t}\right\} $.
The estimated coefficients of the hazard benchmark, which uses the complementary log-log link function and fixed effects for $\xi_{z}$ and $\eta_{t}$, are provided in Table~\ref{tab:hazardCoeffs} (we omit the fixed-effect estimates).
\begin{table}[H]
\begin{center}
\caption{Estimated coefficients of the hazard benchmark model (b).\label{tab:hazardCoeffs}}
{\scriptsize
\begin{tabular}{lrrr}
\toprule
Coefficients & Estimate & S.E. & p-value\bigstrut\\
\midrule
Intercept & $-7.23$ & $1.61$ & $7.34\times10^{-6}$\\
$\text{NPV}_{it}$ & $2.15\times10^{-4}$ & $0.12\times10^{-4}$ & $\simeq0$\\
$\text{NPV}_{H_{it}}$ & $-9.19\times10^{-6}$ & $1.08\times10^{-6}$ & $\simeq0$\\
$H_{it}$ & $3.57\times10^{-2}$ & $0.18\times10^{-2}$ & $\simeq0$\\
$\text{FootPrint2TreeCover}_{i}$ & $2.93\times10^{-4}$ & $0.72\times10^{-4}$ & $4.81\times10^{-5}$\\
$\text{SunlightReceived}_{i}$ & $4.47\times10^{-4}$ & $0.64\times10^{-4}$ & $2.71\times10^{-12}$\\
$\text{Hillshade}_{i}$ & $1.76\times10^{-2}$ & $0.84\times10^{-2}$ & $0.036$\\
$\text{SolarIrradiance}_{i}$ & $-0.41\times10^{-2}$ & $0.18\times10^{-2}$ & $0.027$\\
\bottomrule
\end{tabular}
}
\end{center}
\end{table}
\subsection*{Benchmark (b): Myopic choice model (static NPV logit)}
\begin{description}
\item[]Utility for adopting:
$u_{i1t}(c_t,\bmh_{it},\xi_{i},\varepsilon_{i1t};\rho, \alpha_{e_i},\bm\gamma_{g_i})= \rho + \alpha_{e_i} \text{NPV}_{it}(c_t) +\bm\gamma_{g_i}'\bmh_{it} + \xi_{i}+\varepsilon_{i1t}$.
\item[]Utility for not adopting: $u_{i0t}(\varepsilon_{i0t}) = \varepsilon_{i0t}$.
\end{description}
Then, at iteration $(r)$ of the MCMC algorithm, the probability that household $i$ adopts in period $t$ is:
\begin{align*}
\hat p^{(r)}_{i1t}(c_t,\bm{h}_{it}, \xi^{(r)}_{i};\bm\theta^{(r)}) &= \frac{\exp u^{(r)}_{i1t} \left({c}_{t},\bm{h}_{it},\xi^{(r)}_{i}; \bm\theta^{(r)}\right)}{ 1 +\exp u^{(r)}_{i1t}\left({c}_{t},\bm{h}_{it},\xi^{(r)}_{i} ; \bm\theta^{(r)}\right)}.
\end{align*}
\subsection*{Benchmark (c): Partially forward-looking model}
This model assumes that each household $i$ at period $t$ projects the neighborhood installed base as constant over the planning horizon, i.e., $\bmh_{i,t+1} = \bmh_{it}$, while remaining forward-looking about the net present value of its own system, forecasting the future net cost with equation (7) of the paper.
\subsection*{Benchmark (d): Bass diffusion model (aggregate data)}
We estimated a classic Bass model at the aggregate level, and we projected the number of adopters in out-of-sample quarters. Let $N_t$ be the number of adopters in period $t$, $M$ the market size, $p$ the coefficient of innovation, and $q$ the coefficient of imitation. The regression to estimate is:
\begin{align}
N_t &= M \frac{(p+q)^2}{p} \frac{\exp(-(p+q)t)}{\left(1 + \frac{q}{p} \exp(-(p+q)t)\right)^2}\nonumber.
\end{align}
We use a nonlinear least squares regression (R software, nls \{stats\} function) to estimate the parameters of this model. We obtain $p = 0.0005006$ and $q = 0.0464439$. The out-of-sample MAE is 106.172 adopters and the MAPE is 47.208\%. The Bass model performs poorly because the diffusion process in this dataset is at an early stage and does not yet attain the \textit{S}-shape that the Bass equation fits.
Table~\ref{tab:ECbenchmarks} below reports the out-of-sample MAE and MAPE for the benchmark models (a)--(d) and Model~(4)H of Section~5.1 of the paper.
\begin{table}[H]
\begin{center}
\caption{Out-of-sample predictive performance of the benchmark models.\label{tab:ECbenchmarks}}
{\footnotesize
\begin{tabular}{clcc}
\toprule
\textbf{Model} & \textbf{Specification} & \multicolumn{1}{c}{\textbf{MAE}} & \multicolumn{1}{c}{\textbf{MAPE [\%]}}\bigstrut\\
\midrule
(4)H & Dynamic discrete choice model (best model) & 31.845 & 15.599 \bigstrut\\
(a) & Hazard model (individual level) & 41.453 & 21.831 \\
(b) & Myopic choice model (static NPV logit) & 78.158 & 43.376 \\
(c) & Partially forward-looking model & 48.167 & 26.352 \\
(d) & Bass diffusion model (aggregate data) & 106.172 & 47.208 \\
\bottomrule
\end{tabular}\\
}
\end{center}
\vspace{0.2cm}
{\footnotesize
\setlength{\baselineskip}{10pt}
\noindent \emph{Note.} MAE, mean absolute error (number of adopters); MAPE, mean absolute percentage error.Metrics are computed on the five out-of-sample quarters following the 33-quarter training window.
\par}
\end{table}
\clearpage
\section{Details of Differentiated Rebate Policy Simulations}\label{sec:ECtargeting}
\textbf{Geographic differentiation.} We evaluate rebate policies that differentiate the rebate rate across the geographic segments through a complete factorial: the net cost faced by urban, suburban, and rural households is each scaled by a multiplier in $\{0.8, 1.0, 1.2\}$, for $3^3 = 27$ policies; a lower multiplier corresponds to a higher rebate rate. Each policy is one forward simulation of the selected model, Model~(4)H, under a common rebate schedule. The simulation seeds on the first 16 quarters of observed adoption and generates new adoption from quarter~17 onward, so the reported horizon, period~56, corresponds to 10 years of simulated adoption. For each policy we record the cumulative adoption and the program budget at period~56. Figure~7 in the main text plots cumulative adoption against program budget for the 27 policies, the three-square glyph at each point encoding the urban, suburban, and rural multiplier (left, center, and right square) by color; Table~\ref{tab:ECtargeting} summarizes the results as regressions across the 27 policies.
Cumulative adoption rises almost linearly with the program budget, from \$38.4~million and 7{,}837 adopters at the most conservative policy to \$45.0~million and 8{,}480 adopters at the most generous: column~(1) of Table~\ref{tab:ECtargeting} regresses adoption on budget across the 27 policies and gives a slope of 94.9 adopters per million dollars with $R^2 = 0.97$. Column~(2) regresses adoption on the three multipliers directly: lowering any segment's net cost raises total adoption, with the suburban multiplier the strongest lever (75 additional adopters per $0.1$ discount, against 54 for urban and 31 for rural). Column~(3) asks the budget-constrained question by regressing each policy's deviation from the fitted line of column~(1) on the multipliers, and the urban multiplier separates the policies: a $0.1$ urban discount adds about 14 adopters beyond what its budget predicts, whereas suburban and rural discounts move a policy below the line. Accordingly, all nine policies that discount the urban net cost (multiplier $0.8$) lie above the fitted line, by $+13$ to $+52$ adopters ($+32$ on average), whereas the policies that hold the urban net cost at its observed level or increase it lie below the line on average ($-8$ and $-25$ adopters, respectively). This is the pattern visible in Figure~7 of the main text: the glyphs above the fitted relationship have the left (urban) square in green.
The worked examples in Section~6.4 of the paper come from the same simulations. At a budget of about \$41~million, the policy $(0.8, 1.2, 1.0)$ attains the highest adoption in its budget band, 8{,}120 adopters at \$40.87~million. At a budget of about \$42~million, the policy $(0.8, 1.0, 1.2)$ attains 8{,}212 adopters at \$41.90~million, against 8{,}154 adopters at \$41.79~million under the observed uniform policy $(1.0, 1.0, 1.0)$.
\begin{table}[H]
\begin{center}
\caption{Regression summary of the 27 geographically differentiated rebate policies at period~56. \label{tab:ECtargeting}}
\centering{\footnotesize
\setlength{\tabcolsep}{8pt}\begin{tabular}{lrrr}
\toprule
& (1) & (2) & (3) \\
Dependent variable & Adoption & Adoption & Deviation from budget line \\
\midrule
Budget (\$ million) & 94.86 & & \\
& {\scriptsize (3.15)} & & \\
Urban multiplier $m_{\mathrm{urban}}$ & & $-$544.05 & $-$141.87 \\
& & {\scriptsize (9.21)} & {\scriptsize (7.10)} \\
Suburban multiplier $m_{\mathrm{suburban}}$ & & $-$752.41 & 52.74 \\
& & {\scriptsize (9.21)} & {\scriptsize (7.10)} \\
Rural multiplier $m_{\mathrm{rural}}$ & & $-$310.90 & 41.63 \\
& & {\scriptsize (9.21)} & {\scriptsize (7.10)} \\
Constant & 4,198.34 & 9,764.80 & 47.50 \\
& {\scriptsize (131.64)} & {\scriptsize (16.02)} & {\scriptsize (12.35)} \\
\midrule
$R^2$ & 0.973 & 0.998 & 0.955 \\
Policies & 27 & 27 & 27 \\
\bottomrule
\multicolumn{4}{l}{\scriptsize \emph{Note.} One observation per policy; the uniform policy $(1.0, 1.0, 1.0)$ is the observed rebate schedule.}\\
\end{tabular}}
\end{center}
{\footnotesize
\setlength{\baselineskip}{10pt}
\noindent \emph{Note.} Column~(1) regresses cumulative adoption on the program budget; column~(2) regresses adoption on the three net-cost multipliers; column~(3) regresses the deviation of adoption from the fitted budget line of column~(1) on the multipliers. Standard errors in parentheses. A lower multiplier corresponds to a higher rebate rate, so the negative urban coefficient in column~(3) means that only the urban discount raises adoption beyond what its budget predicts.
\par}
\end{table}
\textbf{Home-value differentiation.} The same design applies to the two economic segments: the net cost of households with home market value below \$300{,}000 and of those at or above it is each scaled by a multiplier in $\{0.8, 1.0, 1.2\}$, for $3^2 = 9$ policies, each a forward simulation of Model~(4)H evaluated at period~56. Figure~\ref{fig:ECeconglyph} plots cumulative adoption against program budget for the nine policies, the two-square glyph at each point encoding the low- and high-value multiplier (left and right square) by color. The uniform policy $(1.0, 1.0)$ reproduces the geographic baseline exactly, 8{,}154 adopters at \$41.79~million, a cross-check between the two simulation sets.
Adoption again tracks the budget almost one for one: regressing adoption on budget across the nine policies gives a slope of 95.7 adopters per million dollars with $R^2 = 0.998$, so the glyphs hug the fitted line and no home-value policy delivers adoption much beyond what its budget predicts. Relative to the uniform policy, discounting the high-value segment's net cost to $0.8$ adds about 246 adopters and discounting the low-value segment's adds about 82, at 99.1 and 111.3 marginal adopters per incremental million dollars, respectively: the segment that adds more adoption also absorbs proportionally more budget. Differentiating the rebate by home value therefore does not substantially improve adoption for a fixed budget, the conclusion reported in Section~6.4 of the paper.
\begin{figure}[H]
{\centering
\includegraphics[width=0.72\textwidth]{fig_econtargeting_glyph.pdf}
\caption{Home-value differentiated rebate policies at period~56.\label{fig:ECeconglyph}}
}
{\footnotesize
\setlength{\baselineskip}{10pt}
\noindent \emph{Note.} Cumulative adoption against program budget for the nine economic-segment policies. The two-square glyph at each point encodes the multiplier on the net cost of low-value households ($[0,300\mathrm{K})$, left square) and high-value households ($[300\mathrm{K},\infty)$, right square): green $0.8$, yellow $1.0$, red $1.2$.
\par}
\end{figure}
\clearpage
\section{End-Horizon Sensitivity Analysis for Policy Simulations}\label{sec:ECrobustness}
The cost-effectiveness comparisons of the rebate switching policies in Section~6.1 are made at period~56, 23 quarters (nearly 6 years) beyond the estimation window. We test whether those comparisons are robust to the end horizon by reconstructing the frontier analyses at five horizons, periods 40, 44, 48, 52, and 56 at four-period steps, equal to 2 through 6 years beyond the last quarter used for estimation (quarter~33). Because a one-step policy RSP($\tau$) that switches at $\tau>T$ does not switch before the end horizon $T$ and hence coincides with the no-switch schedule, we evaluate at each horizon only the policies that switch within it, $\tau\le T$. Figure~\ref{fig:ECrobustness} reconstructs the frontier at each horizon: cumulative installed capacity against program budget on the left, cumulative carbon savings against program budget on the right; Table~\ref{tab:ECrobustness} lists the carbon-maximizing policy and its metrics at each horizon.
Both features of the period-56 frontier reappear across the horizons. Installed capacity rises with budget as a concave curve, so each later switch yields less capacity gain and the costliest policies occupy the flat upper arm. Carbon savings are non-monotone in budget: at every horizon the curve has an interior maximum, with the switches before and after the optimum both below it. At every horizon, the carbon-maximizing one-step policy is an interior switch, never the most expensive policy: RSP(32) at the 2-year horizon and RSP(36) from 3 years onward. The budget at the optimum is about \$30~million at 2 years and about \$33~million from 3 years on, while the carbon level grows from 174.2 at 2 years to 604.2 at 6 years.
The key reason for the interior optimum is the forward-looking behavior of households. Because households know the rebate is reduced at $\tau$, an earlier switch concentrates installations in the quarters before it; that capacity operates sooner and accrues carbon savings over more of the horizon. A later switch installs marginally more capacity by the horizon's end, 38.6~MW for RSP(48) against 36.4~MW for RSP(36) at period~56, yet accumulates less carbon, 552.3 against 604.2, because its capacity arrives later. Cumulative carbon savings, computed as the integral of installed capacity over time, reward the intermediate switch that front-loads adoption. Each policy's program budget plateaus once its rebate is switched off, so it changes little at horizons beyond the switch; the optimum's budget rises across horizons only because the optimal switch time itself moves later.
The counterfactual conclusion therefore holds independently of the period-56 horizon. Evaluated over any window from 2 to 6 years beyond the estimation window, an intermediate switch yields the most carbon savings per budget dollar, installed capacity exhibits diminishing returns to budget, and deferring the switch past the interior optimum raises spending while lowering cumulative carbon. The cost-effective switch advances from RSP(32) at 2 years to RSP(36) from 3 years onward, and it is interior at every horizon.
\begin{table}[H]
\begin{center}
\caption{Horizon robustness of the one-step cost-effectiveness frontier.\label{tab:ECrobustness}}
{\footnotesize
\begin{tabular}{rrlrrr}
\toprule
Horizon (period) & Years beyond est. & Carbon-max policy & Budget (\$M) & Installed MW & Carbon savings \\
\midrule
40 & 2 & RSP(32) & 29.6 & 15.7 & 174.2 \\
44 & 3 & RSP(36) & 32.6 & 21.8 & 250.8 \\
48 & 4 & RSP(36) & 32.6 & 27.1 & 347.9 \\
52 & 5 & RSP(36) & 32.6 & 32.3 & 466.5 \\
56 & 6 & RSP(36) & 32.6 & 36.4 & 604.2 \\
\bottomrule
\end{tabular}}
\end{center}
\vspace{0.2cm}
{\footnotesize
\setlength{\baselineskip}{10pt}
\noindent \emph{Note.} At each simulation horizon, the carbon-maximizing one-step policy with its program budget, cumulative installed capacity, and cumulative carbon savings (the carbon metric of Figure~5 of the paper). The optimum is RSP(32) at 2 years and RSP(36) from 3 years onward, and at every horizon it is an interior switch; period~56 is the headline horizon, nearly 6 years beyond the estimation window.
\par}
\end{table}
\begin{figure}[H]
{\centering
\includegraphics[width=\textwidth]{OneStepFrontier_HorizonRobustness.pdf}
\caption{One-step cost-effectiveness frontier at five simulation horizons.\label{fig:ECrobustness}}
}
{\footnotesize
\setlength{\baselineskip}{10pt}
\noindent \emph{Note.} Periods 40 to 56 (2 to 6 years beyond the estimation window, whose last quarter is~33). Each row reconstructs the period-56 frontier at one horizon: cumulative installed capacity against program budget (left) and cumulative carbon savings against program budget (right), over the one-step policies RSP($\tau$) that switch within the horizon, $\tau\le T$, each point labeled by $\tau$. Installed capacity is concave in budget at every horizon; carbon savings peak at an interior switch, RSP(32) at 2 years and RSP(36) from 3 years onward.
\par}
\end{figure}
\end{APPENDICES}
\clearpage
\bibliographystyle{informs2014}
\bibliography{library}