\section{Results}

We cover the results sequentially in time, first estimating the response of \ac{CREP} wells to the pumping fee that begins in 2011 in \cref{SEC2011} then estimating the water reductions of \ac{CREP} in \cref{SECCREP}. 

\subsection{2011 Pumping Fee and Subdistrict Policies}
\label{SEC2011}
The results of the \ac{DID} for the pumping rate effect in 2011 are provided in \cref{REG2011}. These results provide strong statistical evidence that wells which eventually enroll in \ac{CREP} have a heightened response to the pumping fee. On average, wells in the subdistrict reduce output by 30.9 \ac{AF} per year while wells in \ac{CREP} reduce groundwater use by 62.0 \ac{AF} per year, more than double that of other wells.  
\newpage
\input{Tables/REG2011.tex}

Given the safeguards intended to prevent entry by farmland with low levels of water use, it is worth discussing how the \ac{CREP} wells could have lower levels of water use than other subdistrict wells. To be able to enter \ac{CREP} at least one-half \ac{AF} must be applied to the cropland for four out of the six years from 2008 until 2013. The pumping fee was \$45 per \ac{AF} in 2011 and raised to \$75 in 2012. This means that pumping choices optimized with the \$75 fee were only made for two of the six years. Each groundwater user faced higher cost for those two years, but conservation effort benefits and cost can vary across groundwater users due to aquifer, well, and land characteristics \citep{manning2019,rouhirad2020,guilfoos2013,ekpe2021}. Wells that became sub-economic to operate due to the higher pumping fee in 2012 can still be enrolled in \ac{CREP} since, prior to this, they pumped more than the average. According to the policy criteria, these wells are eligible for enrollment even though there is a reasonable expectation that the reduction in water use is a permanent shift caused by new pumping costs.

The effectiveness of the pumping fee interacts with the \ac{CREP} effect, significantly reducing the net gains of the program. Taking the assumption that each \ac{CREP} well would have ended up under the same steady state water use, with or without the subdistrict policies, then the 62.0 \ac{AF} per year reduction is lost from the \ac{CREP} program. This assumption is within reason, the \ac{CREP} program requires the land be managed in a particular way, only allowing cover crop to be planted. The water required to maintain this type of fallowing is independent of historic pumping rates, so each \ac{CREP} well will reach the same final steady state. 

Past research has identified pumping fees as being a more cost-efficient way to manage water than paying farmers to fallow \citep{rosenberg2020,hendricks2012}. This is a unique case where the water savings identified from \ac{CREP} enrollment can be estimated while a pumping fee is in place. It is in fact the effectiveness of the pumping fee that lowers the effectiveness of \ac{CREP}. If subdistrict wells had not responded to the pumping fee through significant reductions, then the \ac{CREP} program would have a large direct effect on conservation. This highlights the need for policymakers to not only consider the well-established benefits of conservation programs, but to include a broader policy overlap.

These results also highlight the risk of overestimating policy gains by applying historic trends to the enrolled wells. In this case, there was a clear natural experiment where a policy shift created a sudden and widespread change in pumping costs. This makes the selection effect easy to identify empirically. However, this selection effect can be present even without a policy change. Whenever there are unobservable local cost changes then farms that were already on a trajectory to reduce water are more likely to enter the program. Any number of scenarios could arise that cause this local variation in costs. Opportunity costs may rise due to changes in input prices or alternative uses of land. Factors such as urbanization, soil quality, and changes to water stock have been found to significantly affect the enrollment into \ac{CREP} and \ac{CRP} \citep{parks1997,suter2008}. These factors can change at a farm level and on a yearly level. Farms that face higher production costs within two years of the program's start are more likely to enroll into \ac{CREP}, all else equal. In other settings there has been evidence of \ac{CREP} wells lowering water output before enrollment takes place \citep{rosenberg2020}. While these are much milder reduction than is seen in the \ac{SLV}, this farm level selection effect could explain the declines.

Another policy factor worth considering is that the \ac{CREP} program does benefit farmers who face extreme adversity due to pumping costs. Even though the pumping fee is effective at reducing water use, there are uneven distributions of costs and equity concerns \citep{grabenstein2022,ekpe2021}. The selection effect found also means that farms bearing the highest cost of water reduction receive some compensation. The \ac{CREP} program allows for an off-ramp from farming that lets farm owners receive compensation for foregone returns. It would be difficult to craft a payment scheme that can equitably distribute funds based on the costs of the pumping fee. Self-reporting would lead to overestimation of damages, and paying farmers based on the fee paid would undo the fee. Farmers who enter \ac{CREP} must give up producing the land, and so the revealed preferences indicate that they were more affected by the program than other subdistrict users. Since they must fallow the land to enter \ac{CREP}, this compensation does not undo the pumping fee effects on groundwater extraction. While not a stated goal of \ac{CREP}, the compensation of farmers in this manner can even out the costs of the water reduction program.

\subsection{CREP Effects}
\label{SECCREP}

Turning to the effects of the \ac{CREP} program, the estimates from \cref{EQ:SUNAB} are provided in \cref{REGCREP}, with yearly estimates provided graphically in \cref{FIG:EVENTCREP} and \cref{FIG:EVENTNEAR}\footnote{Regression results used to create these figures are provided in \cref{A_CREP_ALL_REG}.}.
\input{Tables/REG_CREP.tex}
The \ac{ATT} of wells that enroll in \ac{CREP} is a reduction of 38.7 \ac{AF} per year. This implies that only 38.4\% of the total well reductions are attributable to entering \ac{CREP}, with the other 61.6\% of reductions attributable to the subdistrict policies. \cref{FIG:EVENTCREP} presents the results as a response across time. There is an immediate reduction of groundwater use of 10 \ac{AF} in year one, but the major reductions do not occur until the second year of the program. Much of the initial program costs are subsidized through the \ac{FSA}, including the cost of planting new native crop cover. This one-year delay reflects the higher necessary water use as farmers are transferred to sustainable fallowing practices. The overall policy response remains large but does drift over time. 


This can be explained in a few ways. First, as the subdistrict increases conservation efforts the difference between the subdistrict wells and the \ac{CREP} wells decreases. If the State of Colorado shut down all wells not enrolled in \ac{CREP} then the program would actually increase groundwater use, as \ac{CREP} wells can still apply small amounts of water to maintain crop cover. A second possibility is that farmers in \ac{CREP} are reallocating groundwater over time. While the rights must be retired on the field, adjacent fields may apply the groundwater from the \ac{CREP} wells to meet their appropriated water volume. We do not distinguish between these possibilities, but in either case the volume of water from the \ac{CREP} wells is maintained well below the counterfactual with the net volume of water dropping by over 95\% of historic levels.

\begin{figure}
    \includegraphics[width=0.95\textwidth]{Figures/CREP_EVENT_STUDY.pdf}
    \caption{Event study of \ac{CREP} wells}
    \label{FIG:EVENTCREP}
\end{figure}

Similar to \cite{rouhirad2021} evaluating \ac{CREP} in Kansas, we are able to identify a policy effect of \ac{CREP} causing neighboring wells to reduce output of pumping. The estimate of a reduction of 2.79 \ac{AF} per year equates to a 3\% reduction in water use of affected wells\footnote{Assuming the yearly average neighbor pumping rate of 92 \ac{AF} per year.}, with a total effect of 3,928 \ac{AF} per year. This compares with the direct \ac{CREP} effect on 5,960 \ac{AF} per year and 40\% of the overall reduction in groundwater pumping come from the spillover effects. 

However, this estimate is an average over the entire \ac{CREP} program, but there is a dynamic component to the reduction outcome. \cref{FIG:EVENTNEAR} presents the neighborhood effects as an event study with adjustments in policy effects over time. Just as in the Kansas \ac{CREP} program, there is a clear decay of \ac{CREP} response with the estimate being statistically insignificant from zero after five years.

\begin{figure}
    \centering
    \includegraphics[width=0.95\textwidth]{Figures/CLOSE_CREP_EVENT_STUDY.pdf}
    \caption{Event study of neighboring \ac{CREP} wells}
    \label{FIG:EVENTNEAR}
\end{figure}

The direction of the effect on neighboring well pumping is not knowable a priori. \ac{CREP} retirements lead to higher water table levels, which in turn reduces pumping costs and creates an incentive to pump more water. This \emph{rebound effect} has been explored in groundwater settings and could cause the \ac{CREP} fallowing to increase the pumping rate of neighbors \citep{jevons1865,pfeiffer2014}. 

Working in the other direction, lowering extraction pressure on a common-pool resource allows the remaining users to manage a large share of the resource, encouraging a Nash equilibrium with jointly lower extraction rates \citep{negri1989,provencher1993,libecap1984}. Another mechanism for lowered neighborhood groundwater use is social norms and ground up informal rule enforcement, leading to a cooperative equilibrium \citep{edwards2021,smith2018,javaid2015,ostrom1989}. The results in \cref{REGCREP} show that neighbors respond in-kind, lowering groundwater use after nearby \ac{CREP} wells stop pumping. This is evidence that the social norms, and cooperative equilibriums dominate these outcomes. However, these pro-social effects interact with the opportunity cost of pumping more water when pumping costs decrease. The decline in neighbor well response as seen in the event study is explained by the increasing opportunity cost of pumping. As the water table rises due to both \ac{CREP} and neighbor wells reduction in pumping, the financial benefit of pumping rises. This leads to users increasing pumping rates on the margin, even if not back to pre-\ac{CREP} levels. This interplay of incentives explains both the initial large decline in well usage and the gradual rebound. 

\subsection{CREP Self-Selection}
Next, the effect of the \ac{SBD1} pumping fee on enrollment numbers in \ac{CREP} is explored. By changing the incentives to farm, the pumping fee can induce additional enrollment in the program. The findings in \cref{SEC2011} and \cref{SECCREP} show that water conservation of each well enrolled in \ac{CREP} is 62\% less due to the prior reductions made to manage costs under the pumping fee. This is only one part of the overall effect of the pumping fee. Since the pumping fee increased fallowing in marginally economic farms, the pumping fee reduces the payment threshold needed to make a \ac{CREP} contract a viable alternative to farming. Farmers decide to enroll land into \ac{CREP} if the opportunity cost of farming is lower than the \ac{PES} amount. By increasing the cost of operating a farm, the fee can lower this opportunity cost, causing some farms to enter \ac{CREP} that would otherwise continue farming. On the other hand, the pumping fee can improve the profitability of farming by controlling externalities. Informed by the pumping statistics, it is argued that reduced water use correlates to reduced farm production and gross revenue. Reducing water as an input to crops means that fewer crops are grown, or less water intensive crops have been substituted. However, the total reduction due to the policy provides an estimate of relative value of water as an input, compared to the externality cost of pumping. Wells that reduce pumping less than the average are able to use an \ac{AF} of water to produce more profit than the typical well. The reduction of water applied after the pumping fee begins can be used to rank the relative costs of the fee to farms. While water intensive farmland that does not shift  water faces higher operating costs, these farms are not near the margin where a \ac{CREP} fee could induce fallowing. For other farms, a large response suggested the fee added a higher cost relative to farm productivity. 

To predict the shift in \ac{CREP} enrollment due to the pumping fee, the reduction of groundwater extraction by a well after the pumping fee begins is used in a probit model. For each well, the average volume of water pumped between 2011 and 2013 is subtracted from the average water extracted in 2009 and 2010. This coefficient indicates if a strong response to the pumping fee drives membership into \ac{CREP}. Ditch fixed effect controls are included to capture the effect of access to surface water, and crop choice variables control for land quality. Crop choices are an indication of the water intensity required to optimize profits prior to the pumping fee implementation, and of soil characteristics. The water rights of wells are included, as possessing more water rights increases the value of a well. Wells providing water to marginally profitable cropland are expected to disproportionately enroll in \ac{CREP}, so water rights can reduce \ac{CREP} entrance. A probit model of \ac{CREP} enrollment is developed in \cref{REG:SELECT_PROBIT}. 

\input{Tables/Probit_mod2.tex}

The direction of each coefficient matches expectations. A larger reduction in water extraction after 2011 makes a well more likely to enter \ac{CREP}. While overall pumping rate after the fee is implemented makes a well less likely to join \ac{CREP}, as shown in model two of \cref{REG:SELECT_PROBIT}. Ownership of water rights is found to decrease the probability of a well joining \ac{CREP}. This variable captures both the effect of access to water rights and well capacity. Data was collected on well pumping tests, but it was not included because the well yield was nearly collinear with water rights. Since water rights are based on historic use, high water rights are strongly correlated with the capacity of wells. Compared to small wells, large wells with more access to water rights are more efficiently employed in areas with high crop density, have lower marginal operating costs, and provide farmers with a stronger legal claim to continue pumping in times of drought. Each of these factors make a well less likely to be a marginal producer that will join \ac{CREP} independent of the pumping fee rate. Similarly, deep wells are less impacted by declines in the water table and correlate to higher capital investment.

The percentage of cropland applied to potatoes prior to the subdistrict formation is also found to significantly decrease the probability of a well entering \ac{CREP}. The excluded fixed effect in the model is \emph{small grains} which is primarily barley within \ac{SBD1}. Compared to small grains, potatoes are drought intolerant and require a more precise soil mineral content \citep{rosen2021}. Furthermore, \ac{SLV} is a major producer of potatoes accounting for 90\% of all potatoes produced in Colorado \citep{nationalagriculturalstatisticsservice2019a}. In 2011, the year the pumping fee began, average gross revenues per acre in the \ac{SLV} were \$4,165 for potatoes\footnote{Assuming a yield of \(\frac{375 cut}{Acre}\) \citep{nationalagriculturalstatisticsservice2019a} and a price of \$9.2 per cut \citep{nationalagriculturalstatisticsservice2013}.} and \$2,943 for barley\footnote{Assuming 115 \(\frac{Bushels}{Acre}\) \citep{nationalagriculturalstatisticsservice2012} at a price of \$25.59 per bushel \citep{internationalmonetaryfund2024}.}. Because there is a higher potential profit per acre of potatoes, higher quality parcels are more likely to grow these crops. While the profitability of small grain farmers and potato farms cannot be compared based on revenues alone, farmland with the lowest operating costs are best used for growing high yield crops. Small grains can be grown where potatoes are planted, but the reverse is not true. The fact that potatoes are selected as a crop indicates that the soil is amenable to producing the higher net value crop. In turn, potato farms are less likely to enter \ac{CREP}.

One of the drivers of this selection effect can be demonstrated using the results from Chapter I. The hedonic model of farmland identifies that crop choice is predictive of the value of land. As the cost of operating a well increases with respect to the pumping fee sequentially higher yield land is taken out of production. Further diseconomies of scale are identified so small segments of large plots tend to be retired before small parcels.

Using the hedonic model each farm parcel in \ac{SBD1} is assigned a per acre land value. The retirement path of crop land in \ac{SBD1} is assessed by removing the lowest marginal parcel from the production in five acre increments. Once a segment of land is removed the marginal value of the remaining land in the parcel increases. The removal of five acre plots is repeated until all land is dropped from production. This is used to present a plausible retirement path of land, showing how the mix of crop changes along this path in \cref{FIG:FEE_CHNG}.
\begin{figure}
    \includegraphics[width=\textwidth]{Figures/IRR_ACRES_EST_CHNG.pdf}
    \caption{Expected retirements from pumping fees}
    \label{FIG:FEE_CHNG}
\end{figure}
As more land is retired, the ratio of water-intensive potatoes changes relative to other crops. A heuristic for assessing the pumping fee is to assume that the inflection point of the least valuable land is the average operating cost of growing the crop. Applying this, the effect on crop mix and active farmland can be estimated using the average price of water applied to each acre of land. The vertical lines in \cref{FIG:FEE_CHNG} represent the average cost addition from the respective fee from the point of marginally profitable farms. The pumping fee retires higher rates of small grain and alfalfa, and these plots are enrolled in \ac{CREP} as the payments to fallow are now higher than the expected returns from irrigating the land.


Due to this observed selection effect there is likely spatial clustering of \ac{CREP} wells that changes the total neighborhood pumping effect. Using the predicted change in groundwater use from \cref{REG2011}, the probit model is estimated under the counterfactual that pumping rates do not change after 2011. The expected number of wells enrolled in \ac{CREP} decreases by 26.07\%, from 154 to 122 wells. The \ac{SBD1} pumping fee induces enrollment into the \ac{CREP}, thereby increasing total water savings. Combining the effect of reduced per well savings and the increase in enrollment, the overall \ac{CREP} water savings are estimated to be 32\% lower than the counterfactual. Compared to the counterfactual, 29.5\% more wells are added to the program. While conservation is increased by this inducement effect the program costs rise by the number of wells added because of the fee, while net conservation is lower.

From the findings in \cref{FIG:EVENTNEAR}, the addition of new wells in \ac{CREP} can create additional neighborhood spillover effects worth considering in the overall policy effectiveness. The addition of 32 new wells in the program provides additional neighborhood effects, but this adjustment changes in a nonlinear way. Two factors contribute to the non-linearity. First, \ac{CREP} wells are not randomly distributed across the subdistrict. For example, potato farmers are less likely to join \ac{CREP}, and land that is ideal for growing potatoes is more alkaline than alfalfa. It follows that CREP enrollment is affected by soil acidity which leads to spatial clustering. Second, even if \ac{CREP} was perfectly random across the subdistrict then the number of untreated wells (farther than a half mile from \ac{CREP}) declines with \ac{CREP} enrollment. On one extreme, if every well in the subdistrict was adjacent to a \ac{CREP} well, adding one more well to \ac{CREP} will not change the number of wells near a well enrolled in fallowing. This means there is a declining marginal spillover effects across \ac{CREP} enrollment. The first well enrolled induces more spillover effect than the final well.

Because the effect depends on regional attributes, and previous enrollment level, the coefficient cannot be applied to estimate this indirect effect. Instead, a Monte Carlo simulation was run using the probit results. For this process, each \ac{CREP} well is randomly assigned a value between zero and one. Then, the predicted probability of each well being in \ac{CREP} is estimated using the results from \cref{REG:SELECT_PROBIT}. This probability is subtracted by the randomly generated number, and then wells are ranked from largest to smallest. For the baseline results, 32 wells are removed from the sample. Then a distance matrix between the wells is used to calculate the counterfactual number of nearby wells. This is repeated 10,000 times to acquire the expected marginal effect on the number of neighboring wells due to the pumping fee inducement of 32 additional \ac{CREP} wells. This leads to an estimate of a 3.27\% increase in the number of wells nearby \ac{CREP}. These results demonstrate the importance of accounting for local characteristics when estimating spillover effects from a \ac{PES} program. Due to the clustering of wells likely to join \ac{CREP}, the 29.5\% increase in enrollment leads to only a minor change in spillover estimates. A policy implication is that the payment for enrollment should vary based on current enrollment. When accounting for spillover effects, the addition of a unit in a \ac{PES} program with many neighbors is more valuable than an addition near other program participants. 

To gain a better idea of how the nearby effects evolve with enrollment, the Monte Carlo is repeated assuming different starting enrollment levels of \ac{CREP}. This process is displayed in \cref{FIG:BOX} as a box plot. As enrollment increases, the rate of change of the number of added neighboring wells declines. If \ac{SBD1} had different characteristics, and was expected to have low enrollment absent intervention, the pumping fee would have had a more significant spillover effect than is estimated. For example, if only 14 wells were expected to enroll in \ac{CREP} prior to the pumping fee, the same addition of 32 wells would increase the number of neighboring wells by over 200. However, the actual addition of 32 wells is expected to add only 40 neighboring wells.


\begin{figure}[!htp]
    \includegraphics[width=0.95\textwidth]{Figures/BOX_PLOT.pdf}
    \caption{Number of wells within a half mile of \ac{CREP} based on total enrollment}
    \label{FIG:BOX}
\end{figure}

\cref{FIG:BOX} also demonstrates that the variance of outcomes is dependent on the number of enrolled wells. The general trend is for the spread of outcomes to converge as \ac{CREP} wells are added. However, low enrollment rates have less variance in outcome than moderate enrollment. When enrollment rates are low, any new addition is likely to pick up some new neighbors. As enrollment increases there is a higher chance that new additions to the program will be near existing \ac{CREP} wells, creating a downwards outlier. However, as enrollment becomes high, it becomes unlikely that a new addition will add any new wells creating a consistent set of outcomes. This evolution of variance is tracked through the box plot whiskers. 

These results are compared to a counterfactual of random enrollment in \ac{CREP} shown in \cref{FIG:BOX_RAND}. This counterfactual presents the expected number of wells that would receive a neighborhood spillover effect from \ac{CREP} if wells were not enrolled in the program based on physical characteristics, or response to the pumping fee but instead were randomly enrolled. This removes the spatial clustering of \ac{CREP} wells increasing the total treatment effect. Compared to a random selection process, the spatial clustering of enrollment minimizes the conservation induced by neighborhood effects.
\begin{figure}[!htp]
    \includegraphics[width=0.95\textwidth]{Figures/BOX_PLOT_WITH_RANDOM.pdf}
    \caption{Number of wells within a half mile of \ac{CREP} when randomly enrolled}
    \label{FIG:BOX_RAND}
\end{figure}


\subsection{Contract Length}
Finally, the effect of varying the length of \ac{CREP} fallowing contracts is estimated. For neighboring wells, the extractive equilibrium may change along the margin of contract length. Of importance for policymakers, changing the terms of the contract will change which farmland is brought into the program. Adjusting the contract length is one way to home in on the most cost-effective allocation of \ac{CREP} payments. \cref{MAINREGTBL} presents the results of estimating \cref{EQ:SUNAB} for wells tied to land that is entered into a permanent contract,  a 15-year \ac{CREP} contract, or a 4-year subdistrict fallowing contract. This is estimated for both the direct effect on wells in the retirement program and for neighboring wells within one-half mile. 
\input{Tables/REG_ALL.tex}
The direct effect of fallowing programs is to reduce groundwater extraction in associated wells. However, the permanently retired contracts are linked to wells with much lower reduction in pumping rates than the shorter 15-year and 4-year terms. This is driven primarily by the well's fixed effects which have different means in each group. Prior to 2011, the average yearly extraction rate was 64.9 \ac{AF} for wells that enroll in the permanent contract, 175 \ac{AF} for wells enrolled in the temporary contract, and 133 \ac{AF} for wells that enroll in the 4-year program. This integrates with the \ac{CREP} literature in multiple areas. First, the short-term contracts do attract wells with a higher-than-expected pumping rate based on entry requirements, as has been found in other settings \citep{rosenberg2020}. However, the permanently retired wells have a pre-policy extraction rate much lower than the average. In survey settings a preference for short-term conservation contracts were found that go beyond expected time value of money considerations \citep{yeboah2015}.  Theoretical models of \ac{CREP} incentives conclude that abatement costs increase when irreversibility is added to the \ac{CREP} terms \citep{yang2004}. The present analysis provides empirical justification that long-term contracts induce entrance by users that differ in economic incentives from those that enter short-term contracts. The fully irreversible contract was only entered into by farms that relied on wells with low productivity.

This result is intuitive given the options available to farmers. There are long-run uncertainties about crop prices, input prices and legal threats. Fields that use large amounts of groundwater to grow crops have a disproportionately large added cost due to the pumping fees. However, such wells also have a larger uncertainty cost of being retired. If prices for crops which require a large volume of water\footnote{Such as potatoes.} rise, then profits of these fields will also increase. Under these uncertainties, permanent land retirement bears a larger cost to high-rate wells. In the short run the cost structure is more well known, and the risk of foregone profits is lower than over the long run. It is not a surprise then that the \ac{CREP} fee can induce large wells to enter into the short-term contracts when faced with higher water extraction costs, but not to enter into contracts that eliminate the option of ever restarting production. The irreversibility of the permanent contract matters more to farms that could foreseeably begin producing water rich crops in the future.

The contract length may also affect the response of neighbor wells, although these results are less robust than the direct effects. The estimated water savings decrease along the margin of contract length. This is consistent with the theory that well neighbors optimize based on the expected game theoretic outcomes. If a field is permanently retired, neighbors can be assured that there will not be a rebound effect when the well enters back into production. More of the common-pool resource is captured by the remaining neighbor wells and a lower pumping rate equilibrium is achievable. In the short-term contracts of four years, neighboring well owners cannot rely on rules of \ac{CREP} to ensure a long-term non-prisoner dilemma outcome. The rebound effect may dominate at these shorter contract lengths since the higher water table provides an incentive to pump now, and nearby wells expect the tragedy of the commons equilibrium state to return once the well enters production. However, it should be noted that the results of the 15 and 4-year contract neighborhood effects are not robust to changes in the model specification\footnote{The 15-year contract has signs of pretend that suggest there could be a larger neighbor effect when including a year anticipation term.}\textsuperscript{, }\footnote{The four-year contract has noisy residuals that are sensitive to the number of lags the policy starts at. There are only two years of data for this contract type, so it is safer to say that there is no evidence of neighborhood pumping declines than to suggest the positive coefficient is definitive.}.