# Groundwater Usage and Strategic Complements: Part I (Instrumental Variables)

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## Abstract

**:**

## 1. Introduction

#### 1.1. The Importance of Groundwater Usage

#### 1.2. Related Literatures

#### 1.3. Our Contribution

#### 1.4. Structure of the Paper

## 2. The Data

- Summary

- Control variables

#### 2.1. Groundwater in the American Midwest

#### 2.2. Governing Groundwater: Nebraska’s Natural Resources Districts

- The “Upper Big Blue” District

#### 2.3. Empirical Regressors

- Farmer’s groundwater usage

- Neighbors

- Groundwater dynamics

## 3. Estimating Strategic Interactions

#### 3.1. Regression Setup

- Strategic network

- Regression Model

- Hypotheses

**Null hypothesis,**${\mathcal{H}}_{\mathbf{0}}^{\prime}$: Framers’ groundwater-usage decisions are strategic substitutes or do not exhibit any systematic relations.

**Alternative hypothesis,**${\mathcal{H}}_{\mathit{A}}^{\prime}$: Framers’ groundwater-usage decisions are strategic complements.

#### 3.2. Identification

#### 3.3. Testing Procedure

**Assumption**

**A1.**

#### 3.4. Results

- Robustness of IVs and Consistency

## 4. Summary of Results

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Instrumental Variables

- (A1.1)
- The individual-time innovations ${\left({\nu}_{i}^{t}\right)}_{(i,t)\in \mathcal{N}\times \mathcal{T}}$ are i.i.d. with finite absolute $4+{\delta}_{\nu}$ moments for some ${\delta}_{\nu}>0$. Furthermore, $\mathbb{E}\left[{\nu}_{i}^{t}\right]=0$ and $\mathbb{E}\left[{\left({\nu}_{i}^{t}\right)}^{2}\right]={\sigma}_{\nu}^{2}>0$.
- (A1.2)
- The matrices $({\mathit{I}}_{N}-{\beta}_{\mathcal{N}}\mathit{A})$ and $({\mathit{I}}_{N}-\rho \mathit{B})$ are non-singular.

#### Appendix A.1. Derivation of IVs

## Appendix B. Robustness Checks

#### Appendix B.2. Strategic Interactions on Different Network Structures

**Table A1.**Hypothesis test results for ${\mathcal{H}}_{0}^{\prime}$ vs. ${\mathcal{H}}_{A}^{\prime}$.

Hypothesis Test Results for ${\mathcal{H}}_{0}^{\prime}$vs.${\mathcal{H}}_{\mathit{A}}^{\prime}$ | |||
---|---|---|---|

$\mathcal{N}(\mathit{\mu},{\mathit{\sigma}}^{2})$ | KS p-Value | p-Value for ${\mathcal{H}}_{0}^{\prime}$vs.${\mathcal{H}}_{\mathit{A}}^{\prime}$ | |

Neighs ≤ 1 mile | $(0.072,0.009)$ | 0.70 | <0.001 |

Neighs ≤ 3 miles | $(0.234,0.014)$ | 0.70 | <0.001 |

Neighs ≤ 5 miles | $(0.351,0.015)$ | 0.70 | <0.001 |

Neighs ≤ 10 miles | $(0.434,0.013)$ | 0.69 | <0.001 |

Neighs ≤ 20 miles | $(0.29,0.009)$ | 0.69 | <0.001 |

**Table A2.**Micro-level regressions on UBB data. Number of observations = 97,160. $\{1,3,5,10,20\}$ mile(s) corresponds to neighborhood sizes as in Figure 4. Coefficients of control variables are normalized such that $\beta =\phantom{\rule{3.33333pt}{0ex}}$% water${}_{i}^{t}$/+1SD variable.

Dependent Variable: Log (Water${}_{\mathit{i}}^{\mathit{t}}$) | |||||
---|---|---|---|---|---|

(1 Mile) | (3 Miles) | (5 Miles) | (10 Miles) | (20 Miles) | |

Strategic interaction effects | |||||

Log(Neighbors’ water${}_{i}^{t}$) | 0.095 *** | 0.271 *** | 0.395 *** | 0.483 *** | 0.339 *** |

(0.015) | (0.023) | (0.027) | (0.027) | (0.026) | |

Groundwater controls | |||||

Spring GW${}_{i}^{t}$ | −0.063 *** | −0.059 *** | −0.057 *** | −0.055 *** | −0.053 *** |

(0.006) | (0.006) | (0.006) | (0.006) | (0.006) | |

Spring GW${}_{i}^{t}$ − Fall GW${}_{i}^{t-1}$ | 0.059 *** | 0.055 *** | 0.051 *** | 0.047 *** | 0.045 *** |

(0.003) | (0.003) | (0.003) | (0.003) | (0.004) | |

UBB-wide time-level controls | |||||

Spot market price${}^{t}$ | 0.027 *** | 0.02 *** | 0.016 *** | 0.014 *** | 0.024 *** |

(0.003) | (0.003) | (0.003) | (0.003) | (0.004) | |

Electricity${}^{t}$ | −0.172 *** | −0.137 *** | −0.112 *** | −0.095 *** | −0.126 *** |

(0.007) | (0.008) | (0.009) | (0.009) | (0.012) | |

Land rental rates${}_{i}^{t}$ | 0.16 *** | 0.129 *** | 0.107 *** | 0.09 *** | 0.102 *** |

(0.004) | (0.005) | (0.005) | (0.006) | (0.006) | |

Rain${}^{t}$ | −0.242 *** | −0.196 *** | −0.163 *** | −0.141 *** | −0.182 *** |

(0.005) | (0.007) | (0.008) | (0.008) | (0.008) | |

Temperature${}^{t}$ | −0.156 *** | −0.124 *** | −0.102 *** | −0.088 *** | −0.122 *** |

(0.006) | (0.007) | (0.007) | (0.008) | (0.01) | |

Farmer-level and time-level controls | |||||

(Temp.) × (Rain)${}^{t}$ | −0.078 *** | −0.065 *** | −0.055 *** | −0.049 *** | −0.069 *** |

(0.004) | (0.004) | (0.005) | (0.005) | (0.007) | |

(Land-size) × (Rain)${}_{i}^{t}$ | −0.017 *** | −0.017 *** | −0.017 *** | −0.017 *** | −0.017 *** |

(0.002) | (0.002) | (0.002) | (0.002) | (0.002) | |

(Land-size) × (Temp.)${}_{i}^{t}$ | −0.012 *** | −0.012 *** | −0.012 *** | −0.012 *** | −0.012 *** |

(0.002) | (0.002) | (0.002) | (0.002) | (0.002) | |

(Well depth) × (Rain)${}_{i}^{t}$ | −0.011 *** | −0.011 *** | −0.01 *** | −0.009 *** | −0.009 *** |

(0.002) | (0.002) | (0.002) | (0.002) | (0.002) | |

(Well depth) × (Temp.)${}_{i}^{t}$ | 0.006 ** | 0.006 ** | 0.005 ** | 0.006 ** | 0.006 ** |

(0.002) | (0.002) | (0.002) | (0.002) | (0.002) | |

(Land-size) × (Rain) × (Temp.)${}_{i}^{t}$ | −0.016 *** | −0.016 *** | −0.016 *** | −0.016 *** | −0.016 *** |

(0.002) | (0.002) | (0.002) | (0.002) | (0.002) | |

(Well depth) × (Rain) × (Temp.)${}_{i}^{t}$ | −0.019 *** | −0.018 *** | −0.018 *** | −0.017 *** | −0.017 *** |

(0.002) | (0.002) | (0.002) | (0.002) | (0.002) |

## Notes

1 | |

2 | To give an idea of its size: if spread across the US, the HPA would cover all fifty states with 1.5 ft of water (https://www.scientificamerican.com/article/the-ogallala-aquifer/ (access on 7 January 2022)). |

3 | The HPA is considered a renewable CPR in Nebraska, since snowmelt from the Rocky Mountains and annual rainfall are, with moderate levels of pumping, sufficient to sustain groundwater levels. |

4 | We would like to cordially thank Rod DeBuhr, Marie Krausnick, and Scott Snell at the UBB for permission and access to UBB data, as well as conversations that have greatly contributed to our study. |

5 | https://www.nass.usda.gov/Statistics_by_State/Nebraska/index.php (access on 1 September 2022). |

6 | https://www.noaa.gov (access on 1 September 2022) |

7 | We are grateful to Dana Divine and Aaron Young for helping us include this data in our analyses. |

8 | More specifically, we utilize a method called “kriging”; see [86,87] for the seminal texts and [88] for a modern treatment. Under reasonable assumptions, this method provides the best linear unbiased prediction of geo-spatial intermediate values that maintains geologically relevant properties, such as continuity of the water table. |

9 | Note that well depth despite the active decision to drill a well is not per se a choice variable, because wells are drilled as deep as needed to get to the groundwater. |

10 | Transmissivity is a metric for how fast groundwater moves across the groundwater basin. Importantly, as a farmer irrigates, higher transmissivity implies the basin ‘replaces’ faster and alleviates the stress on water table levels during pumping, which helps stabilizes groundwater levels. Hence, farmers with higher transmissivity are less prone to receding water table levels. |

11 | When implementing this regression, we work with $log{w}_{i}^{t}$ rather than ${w}_{i}^{t}$ since ${w}_{i}^{t}<0$ is not possible. We remove log to keep notation lighter. |

12 | See [95] for an overview of the reflection problems and various approaches to resolving it. In spatial/network games à la [96], using IVs is one of the most systematically explored means of resolving endogeneity; see, e.g., [97]. We utilize a technique proposed by [74] because it allows us to incorporate spatially correlated errors. |

13 |

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**Figure 1.**Average water-per-acre usage in the Upper Big Blue District from 2007–2014. One inch = one inch of groundwater on top of one acre of land, which is 27,157 gallons.

**Figure 2.**Well-per-well change in water table levels (i.e., groundwater levels) from Spring 2017–14 in the Upper Big Blue District.

**Figure 3.**Results for KS-test of the distribution of (second column) ${\beta}_{\mathcal{N}}-2\mathrm{SE}$ and (third column) ${\mathcal{H}}_{0}^{\prime}$ vs. ${\mathcal{H}}_{A}^{\prime}$.

**Figure 4.**Distribution of estimated ${\beta}_{\mathcal{N}}$ minus 2SE for randomly sampled network structures. Interpretation of ${\beta}_{\mathcal{N}}$: $+1{w}_{{\mathcal{N}}_{i}}^{t}$ corresponds to $+(100\%\times {\beta}_{\mathcal{N}})$ increase in ${w}_{i}^{t}$.

**Table 1.**Descriptive statistics of groundwater (GW) usage and other variables in the Upper Big Blue District, 2008–2012.

Panel A | (2008) | (2009) | (2010) | (2011) | (2012) |

Number of observations | 10,375 | 10,546 | 10,435 | 10,426 | 10,714 |

GW-usage${}_{i}^{t}$ | 5.79 | 8.27 | 6.38 | 5.85 | 13.81 |

(GW-usage${}_{i}^{t}$ SD) | (4.2) | (5.1) | (4.3) | (4.1) | (6.9) |

Groundwater control variables | |||||

Spring GW${}_{i}^{t}$ (ft) | 81.18 | 81.89 | 81.98 | 81.68 | 82.00 |

(Spring GW${}_{i}^{t}$ SD) | (12.2) | (12.3) | (12.2) | (12.6) | (11.9) |

Fall GW${}_{i}^{t}$ (ft) | 82.89 | 82.67 | 80.70 | 78.47 | 84.58 |

(Fall GW${}_{i}^{t}$ SD) | (9.8) | (8.5) | (8.6) | (8.8) | (7.5) |

Annual control variables | |||||

Price of corn${}^{t}$ ($/bushel) | 5.65 | 3.55 | 3.68 | 7.17 | 7.17 |

Electricity${}^{t}$ (¢/kW-hr) | 11.26 | 11.51 | 11.54 | 11.72 | 11.88 |

Rain${}^{t}$ (in) | 16.24 | 13.85 | 18.54 | 22.33 | 6.52 |

Temperature${}^{t}$ (${}^{\circ}$F) | 71.48 | 68.45 | 71.08 | 71.03 | 72.60 |

Farmland rental rates${}_{C}^{t}$ ($/acre) | 170.5 | 173.5 | 180.6 | 197.9 | 243.8 |

(Rental rate${}_{C}^{t}$–SD) | (8.2) | (8.2) | (8.6) | (9.4) | (18.6) |

Panel B | (mean) | (SD) | (Range) | ||

Farm control variables | |||||

Land-size${}_{i}$ (acres) | 101.51 | 35.72 | [3.0, 349.0] | ||

Well Depth${}_{i}$ (ft) | 184.82 | 31.93 | [5.0, 470.0] | ||

Transmissivity${}_{i}$ (ft${}^{3}$) | 126.34 | 36.25 | [1.66, 249.3] |

Dependent Variable: Log (Water${}_{\mathit{i}}^{\mathit{t}}$) | ||||||
---|---|---|---|---|---|---|

(1) | (2) | (3) | (4) | (5) | (6) | |

Strategic interaction effects | ||||||

Log(Neighbors’ water${}_{i}^{t}$) | 0.956 *** | 0.390 *** | 0.307 | 0.237 *** | 0.948 *** | 0.395 *** |

(0.021) | (0.027) | (0.352) | (0.032) | (0.022) | (0.027) | |

Groundwater controls | ||||||

Spring GW${}_{i}^{t}$ | −0.03 *** | −0.051 *** | −0.034 *** | −0.057 *** | ||

(0.006) | (0.006) | (0.006) | (0.006) | |||

Spring GW${}_{i}^{t}$ − Fall GW${}_{i}^{t-1}$ | 0.019 *** | 0.044 *** | 0.023 *** | 0.051 *** | ||

(0.003) | (0.003) | (0.003) | (0.003) | |||

UBB-level time-dependent controls | ||||||

Spot market price${}^{t}$ | 0.018 *** | 0.034 *** | 0.016 *** | |||

(0.003) | (0.003) | (0.003) | ||||

Electricity${}^{t}$ | −0.115 *** | −0.155 *** | −0.112 *** | |||

(0.009) | (0.01) | (0.009) | ||||

Land rental rates${}_{{C}_{i}}^{t}$ | 0.107 *** | 0.126 *** | 0.107 *** | |||

(0.006) | (0.006) | (0.005) | ||||

Rain${}^{t}$ | −0.164 *** | −0.199 *** | −0.163 *** | |||

(0.008) | (0.009) | (0.008) | ||||

Temperature${}^{t}$ | −0.105 *** | −0.145 *** | −0.102 *** | |||

(0.007) | (0.009) | (0.007) | ||||

(Temp.)$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}$(Rain)${}^{t}$ | −0.055 *** | −0.061 *** | −0.055 *** | |||

(0.005) | (0.005) | (0.005) | ||||

Farmer- and time-dependent controls | ||||||

(Land-size)$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}$(Rain)${}_{i}^{t}$ | −0.017 *** | −0.016 *** | −0.017 *** | −0.017 *** | ||

(0.002) | (0.002) | (0.002) | (0.002) | |||

(Land-size)$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}$(Rain)${}_{i}^{t}$ | −0.012 *** | −0.012 *** | −0.012 *** | −0.012 *** | ||

(0.002) | (0.002) | (0.002) | (0.002) | |||

(Well depth)$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}$(Rain)${}_{i}^{t}$ | −0.007 *** | −0.007 *** | −0.008 *** | −0.01 *** | ||

(0.002) | (0.002) | (0.002) | (0.002) | |||

(Well depth)$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}$(Temp.)${}_{i}^{t}$ | 0.005 ** | 0.006 ** | 0.005 ** | 0.005 ** | ||

(0.002) | (0.002) | (0.002) | (0.002) | |||

(Land-size)$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}$(Rain)$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}$(Temp.)${}_{i}^{t}$ | −0.016 *** | −0.015 *** | −0.016 *** | −0.016 *** | ||

(0.002) | (0.002) | (0.002) | (0.002) | |||

(Well depth)$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}$(Rain)$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}$(Temp.)${}_{i}^{t}$ | −0.014 *** | −0.014 *** | −0.016 *** | −0.018 *** | ||

(0.002) | (0.002) | (0.002) | (0.002) |

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**MDPI and ACS Style**

Koch, C.M.; Nax, H.H.
Groundwater Usage and Strategic Complements: Part I (Instrumental Variables). *Games* **2022**, *13*, 67.
https://doi.org/10.3390/g13050067

**AMA Style**

Koch CM, Nax HH.
Groundwater Usage and Strategic Complements: Part I (Instrumental Variables). *Games*. 2022; 13(5):67.
https://doi.org/10.3390/g13050067

**Chicago/Turabian Style**

Koch, Caleb M., and Heinrich H. Nax.
2022. "Groundwater Usage and Strategic Complements: Part I (Instrumental Variables)" *Games* 13, no. 5: 67.
https://doi.org/10.3390/g13050067