\section{Strategy}

\subsection{Pumping Fee and Subdistrict Policies} 
To assess the outcome of the subdistrict's water conservation efforts, a \ac{DID} applying well fixed effects is employed. In this model, wells that join \ac{CREP} are treated as having a different response to the subdistrict policies than other wells in \ac{SBD1}. The \Ac{SBD1} policy interactions with the \ac{CREP} payments are of interest. Heterogeneous response to the pumping fee has been identified in the subdistrict \citep{ekpe2021}, this could change the effectiveness of \ac{CREP}. If the wells that enter \ac{CREP} faced the highest cost from the pumping fee, they may have already reduced water use prior to entering \ac{CREP}. In one extreme, if the pumping fee already induced the \ac{CREP} wells to shut off then the \ac{CREP} program will not reduce the water use of wells. In such a case, the program would provide welfare recovery to disproportionately affected farmers but would not directly assist in conservation efforts. Through a rebound effect, it is possible that \ac{CREP} wells will pump more than their counterpart subdistrict wells. The model estimates the \ac{ATE} of the pumping fee and other policies on both \ac{CREP} and subdistrict wells. This is expressed as a \ac{DID} model in \cref{EQ:2011DID}.
\begin{equation}
	\label{EQ:2011DID}
	Y_{w,t}=\alpha+\phi_{w}+\eta_{t}+\omega_{d,t}(w)+\gamma_{s,t}(w)+Sbd1 \cdot Post+CREP\cdot Post+\epsilon_{w,t}
\end{equation}
The dependent variable \(Y_{w,t}\) is the volume of water pumped in acre-feet for a well \emph{w} in year \emph{t}. The dummy variable \emph{Sbd1} is one if a well is in \ac{SBD1} and zero otherwise. The dummy variable \emph{CREP} is one if a well has ever been enrolled in \ac{CREP} and zero otherwise. All wells in \ac{CREP} are also in \ac{SBD1}, so the \emph{Sbd1} coefficient is one for all wells where \emph{CREP} is one. The dummy variable \emph{Post} is zero before 2011 and is one from 2011 forward. The interaction between \emph{Sbd1} and \emph{CREP} with \emph{Post} are the estimates of note. This interaction captures the average effect of \ac{SBD1} policies on wells and the \emph{CREP} interaction expresses any heterogeneous impact on \ac{CREP} wells. If \ac{CREP} wells behave similarly to other wells in the subdistrict, then the \emph{CREP} and \emph{Post} interaction will be statistically insignificant from zero, otherwise there is evidence that \ac{CREP} wells were affected differently.

Fixed effects include \(\phi_{w}\) well, \(\eta_{t}\) year,  \(\omega_{d,t}\) ditch-year, and \(\gamma_{s,t}\) subdistrict-year. The suite of fixed effects used removes much of the potentially omitted variable bias by controlling for time and space fixed attributes. The well's fixed effect absorbs any time invariant attributes, such as capacity, well appropriation date, and perforation depth. Importantly, they also remove unobserved spatial components such as permeability or local geologic features. The year fixed effects account for variations that affect all \ac{SLV} farmers. These include crop prices, changes in local non-water input prices, federal and state policies, and generalized rainfall. Subdistrict-year effects further capture variations in individual subdistrict policies. All six subdistricts began employing water reduction policies during the explored time periods, this fixed effect captures such changes without explicit controls for pumping fee rate changes. Finally, the ditch year effects incorporate surface water access changes that impact certain users. Based on the interaction of the yearly snow runoff levels with ditch priority, ditch users will have variations in the accessibility of surface water in a given year. This is true even when accounting for yearly average snowmelt and precipitation captured in the year fixed effect. This ditch interaction accounts for this and other ditch time-varying factors affecting demand for groundwater.

The yearly subdistrict policy effects are presented with an event study design to highlight the yearly changes in the policy suite. The changes to the pumping fee and the addition of other conservation policies suggest that the treatment effect will vary over time. \Cref{EQ:2011DID} is rewritten as an event study in \Cref{EQ:2011EVENT}.
\begin{equation}
	\label{EQ:2011EVENT}
	Y_{w,t}=\alpha+\phi_{w}+\eta_{t}+\omega_{d,t}+\gamma_{s,t}+\sum_{t\not=2010}\left(\beta_{t}\cdot\rho_{w,t}\right)\left[Sbd1+CREP\right]+\epsilon_{w,t}
\end{equation}

Where \(\rho_{w,t}\) is an indicator variable for a well \emph{w} being in year \emph{t}. We refer to the estimates as representing the subdistrict policies in general since there are many policy changes occurring simultaneously. The pumping fee is the spearhead policy, but the investment in well purchases, land expansion, and short-term fallowing all contribute to the subdistrict effect.

\subsection{CREP Choice}
The decision of farmers to enroll their land in \ac{CREP} is treated as a simple comparison between the option values of the land. With land being enrolled if the condition in \cref{EQ:CREPCHOICE} is met.
\begin{equation}
	\label{EQ:CREPCHOICE}
	\theta_{0}+\sum_{t=1}^{T} \frac{\theta_{CREP,t}\cdot A_{i}}{\left(r+1\right)^{t}} \ge \sum_{t=1}^{T} \frac{P_{\gamma}\cdot \gamma_{t}\cdot Q_{t,i}-C_{t,i}}{\left(r+1\right)^{t}}
\end{equation}
\(\theta_{0}\) being the initial sign up bonus payment per acre of \ac{CREP}, \emph{T} is the length of a \ac{CREP} contract, \(\theta_{CREP,t}\) is the yearly payment rate per acre, \(A_{i}\) is the area enrolled in \ac{CREP}, \(\gamma_{t}\) is the crop mix grown in a given year, \(P_{\gamma}\) is the weighted average price of the crop mix, \(C_{t,i}\) is the cost to operate the parcel, and \emph{r} is the discount rate.

It follows from this model that the choice to enroll in \ac{CREP} depends on the relative attributes of soil. These attributes affect the crop choice \(\gamma_{t}\) and total yield \(Q_{t,i}\).  In the probit model of \ac{CREP} selection, pre-policy crop choice is included to account for these parcel quality characteristics. The pumping fee affects the cost of operating \(C_{t,i}\), the magnitude of water reductions from the fee is used to capture the relative cost increase from the pumping fee. 
\subsection{CREP and Spillover Effects}
The possibility of time varying heterogeneous group treatment effects should be considered when selecting an empirical strategy for evaluating the effects of \ac{CREP}. Since \ac{CREP} has staggered treatment periods, using treatment lags will bias the regression unless the cohort responses are identical, and there is no pre-trend \citep{sun2021,callaway2021,borusyak2021,goodman-bacon2021,dechaisemartin2020,gardner2022}. The complex nature of the policy implementations of \ac{SBD1} make this a potential issue. Ideally, if the \ac{CREP} participants were drawn randomly then such confounding interactions would be avoided \citep{athey2022}. This is unlikely to be the case since farmers can opt into the \ac{CREP} program, and the period farmers enter \ac{CREP} is contingent on subdistrict policies. \cref{FIG:CREP_GROUP_CHNG} groups wells that entered \ac{CREP} (treated wells) by the start of the \ac{CREP} contract year. The average pumping rate of these wells is calculated from 2009-2010, the range where no subdistrict policies had been implemented. If there are no group selection effects then the outcome variable (pumping) should be consistent prior to any treatment. 

\FloatBarrier
\begingroup
\begin{figure}[h]
	\centering
	\includegraphics[width=.8\textwidth]{CREP_GROUP_TRENDS.jpeg}
	\caption{Pre-2011 groundwater use of wells grouped by \ac{CREP} start year}
	\label{FIG:CREP_GROUP_CHNG}
	% I THINK IS A CUT/PASTE ERROR- \label{FIG:STOR}
\end{figure}
\endgroup
\FloatBarrier

There are large variations in average pre-pumping fee groundwater extraction rates across treatment cohorts. This suggests that an assumption of homogeneous group-time treatment effects would be violated. This is plausible in this dynamic policy scenario. Wells that entered the program at the initial sign-up would include any wells that are marginally profitable under earlier conditions. These wells entered when there was a higher water table, and when the fee was anywhere from \$45-\$75 per \ac{AF}. In this first wave of sign-ups some wells would have been entered even if there was not a pumping fee, while others were induced by higher prices. The pumping fee was later raised to \$150, at about this time the average pumping rate of wells entered into \ac{CREP} increases. The marginal cost of pumping being raised provides a new set of economic signals for farmers deciding if the \ac{CREP} lump sum is worth foregoing crop production. It is not surprising that the well attributes shift with this policy. Another possibility is that farmers learn from experience about the profitability of \ac{CREP}, and this new knowledge changes the type of farmers willing to enroll. 

Given that the well attributes change over time, the possibility of heterogeneous treatment effects cannot be eliminated. To correct for this, a cohort weighted regression is used that results in an \ac{ATT} outcome \citep{sun2021}. The final equation to be estimated is presented in \cref{EQ:SUNAB}, but \cref{EQ:STG1} is the first step that accounts for covariates. This estimates the \ac{CREP} cohort\footnote{The cohorts in this case are the groups of wells that start a \ac{CREP} contract in a given year, there are unique cohorts from 2014-2021} \ac{ATE}.
\begin{equation}
	\label{EQ:STG1}
	Y_{w,d,t,s}=\alpha+\phi_{w}+\eta_{t}+\omega_{d,t}+\gamma_{s,t}+\sum_{e\not= \infty} \sum_{\ell\not=-1}\delta_{\ell,e}(\theta_{w,e}\cdot \rho_{w,t}^{\ell})+\epsilon_{w,d,t,s}
\end{equation}

Where \(\rho_{w,t}^{\ell}\) is an indicator variable for a well \emph{w} being \(\ell\) periods away from treatment in the year \emph{t}, \(e\in E \) is the treatment cohort in this case the year a well enters \ac{CREP}, \(\theta_{w,e}\) is an indicator variable that is one if a well is in the \ac{CREP} treatment cohort \emph{e}. \(\delta_{\ell,e}\) is the coefficient of interest, being the treatments effect on the cohort \emph{e} with lag \(\ell\). The reference cohort is the never treated group \(e=\infty\), and the reference lag period is one year prior to treatment \(\ell=-1\).

Next, weights are estimated which are used to predict the \ac{ATE} of \ac{CREP} from the cohort coefficients \(\delta_{\ell,e}\). These weights are the sample shares of cohorts in each lag period, as performed in \cref{EQ:STG2}.
\begin{equation}
	\label{EQ:STG2}
	\Phi_{e,\ell}=Pr \left\{E_{w}=e\ |\ E_{w} \in \left[-\ell,T-\ell \right] \right\}
\end{equation}
\Cref{EQ:STG2} calculates the probability that the treatment cohort of a well \(E_{w}\) is in the sample of wells treated after a number of lags \(\ell\). If \(\ell=0\) then this is the probability that the cohort of the well was ever treated, and if \(\ell=-2\) then this is the probability that the cohort of the well was treated in the range of two years prior to the first treatment of any well, and at least two years before the end of the sample period.

With these weights, the coefficients of interests can be calculated with \cref{EQ:SUNAB}.
\begin{equation}
	\label{EQ:SUNAB}
	\widehat{CREP\ ATT}_{g}=\frac{1}{|g|}\sum_{\ell \in g} \sum_{e}\hat{\Phi}_{e,\ell}\cdot\hat{\delta}_{\ell,e}
\end{equation}
Where \emph{g} is the set of all lags \(\ell\). The final equation estimates the \ac{ATT}, by the sum of cohort treatment effects estimated in \cref{EQ:STG1} weighted by the cohort sample share in \cref{EQ:STG2} and scaled by the number of periods in the set \(|\)\emph{g}\(|\). This provides  consistent coefficient estimates under time and group varying treatment effects, as is the case for the \ac{CREP} program.

The previous equations are written with regard to the direct effect of \ac{CREP} on wells that are in the program. However, \cref{EQ:SUNAB} is applicable to neighborhood effects of \ac{CREP}. These spillover effects that capture the neighboring well responses to hydrologic shifts, and social norms driven by \ac{CREP} also utilize this model. In such a case the first treatment period is expressed as the first year that a well was within one-half mile of a well that entered \ac{CREP}. Furthermore, all wells in \ac{CREP} are removed from the dataset to avoid attributing direct \ac{CREP} effects to spatial overlap between \ac{CREP} wells.
