# On the Statistical Distribution of the Nonzero Spatial Autocorrelation Parameter in a Simultaneous Autoregressive Model

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## Abstract

**:**

## 1. Introduction

## 2. Model Specification and Parameter Estimation

## 3. The Sampling Distribution of $\widehat{\mathit{\rho}}$

#### 3.1. The Relationship between $\widehat{\rho}$ and the MC

#### 3.2. The Sampling Variance of $\widehat{\rho}$

#### 3.2.1. The Sampling Variance of $\widehat{\rho}$ at Zero

#### 3.2.2. The Sampling Variance of $\widehat{\rho}$ at Nonzero Values

#### 3.2.3. Simulation Experiments

## 4. A Simulated Zero Spatial Autocorrelation Scenario, and Two Nonzero Empirical Examples

#### 4.1. A Description of the Selected Datasets

#### 4.2. Results and Explanations

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Bivariate Regression Results

**Figure A1.**Bivariate regression results of ${\widehat{\rho}}_{MC}$ as a function of $\widehat{\rho}$ for three 70-by-70 ($n=4900$ ) regular tessellations: (

**a**) square rook adjacency case; (

**b**) square queen adjacency case; and, (

**c**) hexagonal case data.

**Figure A2.**Bivariate regression results of the asymptotic variance of $\widehat{\rho}$ as a function of its theoretical version for three 150-by-150 ($n=22,500$ ) regular tessellations: (

**a**) square rook adjacency case; (

**b**) square queen adjacency case; and, (

**c**) hexagonal case.

## Appendix B. Components of the Denominator of the Asymptotic Variance of $\widehat{\rho}$ at Zero for Different Landscapes

Landscape | ${{\displaystyle \sum}}_{\mathit{i}=1}^{\mathit{n}}1/{{\displaystyle \sum}}_{\mathit{j}=1}^{\mathit{n}}{\mathit{c}}_{\mathit{i}\mathit{j}}$ | ${{\displaystyle \sum}}_{\mathit{i}=1}^{\mathit{n}}{\mathit{\lambda}}_{\mathit{i}}^{2}$ |
---|---|---|

Square rook | $\left[3PQ+2\left(P+Q\right)+4\right]/12$ | $\left[18PQ+11\left(P+Q\right)+12\right]/72$ |

Square queen | $\left[15PQ+18\left(P+Q\right)+28\right]/120$ | $\left[300PQ+279\left(P+Q\right)+234\right]/2400$ |

Hexagonal | $\left(5PQ+5P+6Q+8\right)/30$ | $\left(30PQ+25P+24Q+23\right)/180$ |

## Appendix C. Basic Information about the Simulation Experiments

Regular Square Rook | Regular Square Queen | Regular Hexagonal Tessellation | ||||||
---|---|---|---|---|---|---|---|---|

Sample Size | Number of Methods | Number of Replications | Sample Size | Number of Methods | Number of Replications | Sample Size | Number of Methods | Number of Replications |

10-by-10 | 3 | 3*21*10,000 | 10-by-10 | 3 | 3*29*10,000 | 10-by-10 | 3 | 3*28*10,000 |

15-by-15 | 2 | 2*21*10,000 | 15-by-15 | 2 | 2*29*10,000 | 15-by-15 | 2 | 2*28*10,000 |

20-by-20 | 2 | 2*21*10,000 | 20-by-20 | 2 | 2*29*10,000 | 20-by-20 | 2 | 2*28*10,000 |

25-by-25 | 2 | 2*21*10,000 | 25-by-25 | 1 | 29*10,000 | 25-by-25 | 2 | 2*28*10,000 |

30-by-30 | 2 | 2*21*10,000 | 30-by-30 | 1 | 29*10,000 | 30-by-30 | 2 | 2*28*10,000 |

35-by-35 | 2 | 2*21*10,000 | 35-by-35 | 1 | 29*10,000 | 35-by-35 | 2 | 2*28*10,000 |

40-by-40 | 2 | 2*21*10,000 | 40-by-40 | 1 | 29*10,000 | 40-by-40 | 2 | 2*28*10,000 |

45-by-45 | 2 | 2*21*10,000 | 45-by-45 | 1 | 29*10,000 | 45-by-45 | 1 | 28*10,000 |

50-by-50 | 2 | 2*21*10,000 | 50-by-50 | 1 | 29*10,000 | 50-by-50 | 1 | 28*10,000 |

55-by-55 | 2 | 2*21*10,000 | 55-by-55 | 1 | 29*10,000 | 55-by-55 | 1 | 28*10,000 |

60-by-60 | 2 | 2*21*10,000 | 60-by-60 | 1 | 29*10,000 | 60-by-60 | 1 | 28*10,000 |

65-by-65 | 2 | 2*21*10,000 | 65-by-65 | 1 | 29*10,000 | 65-by-65 | 1 | 28*10,000 |

70-by-70 | 2 | 2*21*10,000 | 70-by-70 | 1 | 29*10,000 | 70-by-70 | 1 | 28*10,000 |

75-by-75 | 2 | 2*21*10,000 | 75-by-75 | 1 | 29*10,000 | 75-by-75 | 1 | 28*10,000 |

80-by-80 | 2 | 2*21*10,000 | 80-by-80 | 1 | 29*10,000 | 80-by-80 | 1 | 28*10,000 |

85-by-85 | 2 | 2*21*10,000 | 85-by-85 | 1 | 29*10,000 | 85-by-85 | 1 | 28*10,000 |

90-by-90 | 2 | 2*21*10,000 | 90-by-90 | 1 | 29*10,000 | 90-by-90 | 1 | 28*10,000 |

95-by-95 | 2 | 2*21*10,000 | 95-by-95 | 1 | 29*10,000 | 95-by-95 | 1 | 28*10,000 |

100-by-100 | 2 | 2*21*10,000 | 100-by-100 | 1 | 29*10,000 | 100-by-100 | 1 | 28*10,000 |

105-by-105 | 2 | 2*21*10,000 | 105-by-105 | 1 | 29*10,000 | 105-by-105 | 1 | 28*10,000 |

110-by-110 | 2 | 2*21*10,000 | 110-by-110 | 1 | 29*10,000 | 110-by-110 | 1 | 28*10,000 |

115-by-115 | 2 | 2*21*10,000 | 115-by-115 | 1 | 29*10,000 | 115-by-115 | 1 | 28*10,000 |

120-by-120 | 2 | 2*21*10,000 | 120-by-120 | 1 | 29*10,000 | 120-by-120 | 1 | 28*10,000 |

125-by-125 | 2 | 2*21*10,000 | 125-by-125 | 1 | 29*10,000 | 125-by-125 | 1 | 28*10,000 |

## Appendix D. Correct Derivation of the Jacobian Term for SAS

## Appendix E. Asymptotic Normality of the Sampling Distribution of the Spatial Autocorrelation (SA) Parameter $\mathit{\rho}$

**Figure A3.**Frequency distributions of weak, moderate, and strong positive SA parameters: (

**a**) square tessellation, rook adjacency; (

**b**) square tessellation, queen adjacency; and, (

**c**) hexagonal adjacency.

## Appendix F. $\mathit{\rho}$ versus the Variance of $\widehat{\mathit{\rho}}$: Simulation Experiment Results for Three Adjacencies

**Figure A4.**Theoretical variances (orange) versus exact variances (red and green) obtained by approximating the Jacobian term and employing new maximum likelihood estimators (MLEs) with different sample sizes for the square tessellation with rook adjacency.

**Figure A5.**Theoretical variances (orange) versus exact variances (red and green) obtained by approximating the Jacobian term and employing new MLEs with different sample sizes for the square tessellation with queen adjacency.

**Figure A6.**Theoretical variances (orange) versus exact variances (red and green) obtained by approximating the Jacobian term and employing new MLEs with different sample sizes for the square tessellation with hexagonal adjacency.

## Appendix G. The Sampling Distribution of the SA Parameter $\mathit{\rho}$ for the Wuhan Census Blocks Dataset

**Figure A7.**Curve fitting plots and bivariate regression plots for Wuhan census blocks data. (

**a1**,

**a2**) The original MC-$\widehat{\rho}$ scatter plot (blue) superimposed on the theoretical MC-$\rho $ curve (red), and a bivariate regression scatter plot for the original and predicted $\rho $; (

**b1**,

**b2**) the original $\widehat{\rho}$ -var($\widehat{\rho}$ ) scatter plot (blue) superimposed on the theoretical $\widehat{\rho}$ -var($\widehat{\rho}$ ) curve (red), and a bivariate regression scatter plot for the original and predicted Var($\widehat{\rho}$ ).

## Appendix H. A Data Analysis for One Simulated and Two Empirical Examples

**Figure A8.**Statistical properties of simulated data and their regression residuals: (

**a**) a histogram of simulated data; (

**b**) a Q-Q plot of the SAR model residuals; and, (

**c**) a scatter plot of residual-predicted values for assessing the homoscedasticity of model residuals.

**Figure A9.**Statistical properties of the transformed young population ratio and its regression residuals: (

**a**) a histogram of the exponential transformed young population ratio; (

**b**) a Q-Q plot of the SAR model residuals; and, (

**c**) a scatter plot of residual-predicted values for assessing the homoscedasticity of model residuals.

**Figure A10.**Improved normality of transformed normalized difference vegetation index (NDVI) and regression residuals: (

**a**) a histogram of the Exponential transformed NDVI; (

**b**) a Q-Q plot of the SAR model residuals; and, (

**c**) a scatter plot of residual-predicted values for assessing the homoscedasticity of model residuals.

## Appendix I. Statistical Power Visualization of Different SA Statistics in a Single Plot

**Figure A11.**Statistical power curves for the MC (green), the GR (red), and the SAR $\rho $ (blue): (

**a**) the 7-by-7 square tessellation, rook adjacency case; (

**b**) the 7-by-7 square tessellation, queen adjacency case; and, (

**c**) the 7-by-7 hexagonal tessellation case.

## References

- Li, H.; Calder, C.A.; Cressie, N. Beyond Moran’s I: Testing for Spatial Dependence Based on the Spatial Autoregressive Model. Geogr. Anal.
**2007**, 39, 357–375. [Google Scholar] [CrossRef] - Robinson, P.M.; Rossi, F. Refined tests for spatial correlation. Econom. Theory
**2015**, 31, 1249–1280. [Google Scholar] [CrossRef] - López, F.; Matilla-García, M.; Mur, J.; Marín, M.R. A non-parametric spatial independence test using symbolic entropy. Reg. Sci. Urban Econ.
**2010**, 40, 106–115. [Google Scholar] [CrossRef][Green Version] - Gotelli, N.J.; Ulrich, W. Statistical challenges in null model analysis. Oikos
**2012**, 121, 171–180. [Google Scholar] [CrossRef] - Griffith, D.A. Spatial statistics: A quantitative geographer’s perspective. Spat. Stat.
**2012**, 1, 3–15. [Google Scholar] [CrossRef] - Bivand, R. Package ‘spdep’. Available online: https://cran.r-project.org/web/packages/spdep/spdep.pdf (accessed on 24 October 2018).
- Hong, Y.; White, H. Asymptotic Distribution Theory for Nonparametric Entropy Measures of Serial Dependence. Econometrica
**2005**, 73, 837–901. [Google Scholar] [CrossRef][Green Version] - Matilla-García, M.; Marín, M.R. A non-parametric independence test using permutation entropy. J. Econom.
**2008**, 144, 139–155. [Google Scholar] [CrossRef][Green Version] - Montgomery, D.C.; Peck, E.A.; Vining, G.G. Introduction to Linear Regression Analysis; Wiley: Hoboken, NJ, USA, 2012. [Google Scholar]
- Burt, J.E.; Barber, G.M.; Rigby, D.L. Elementary Statistics for Geographers; Guilford Publications: New York, NY, USA, 2009. [Google Scholar]
- Bewick, V.; Cheek, L.; Ball, J. Statistics review 7: Correlation and regression. Crit. Care
**2003**, 7, 451–459. [Google Scholar] [CrossRef] - Provost, B.S. Closed-Form Representations of the Density Function and Integer Moments of the Sample Correlation Coefficient. Axioms
**2015**, 4, 268–274. [Google Scholar] [CrossRef] - Ames, E.; Reiter, S. Distributions of Correlation Coefficients in Economic Time Series. J. Am. Stat. Assoc.
**1961**, 56, 637–656. [Google Scholar] [CrossRef] - Lee, M.Y. The Effect of Nonzero Autocorrelation Coefficients on the Distributions of Durbin-Watson Test Estimator: Three Autoregressive Models. Expert J. Econ.
**2014**, 2, 85–99. [Google Scholar] - Mead, R. A Mathematical Model for the Estimation of Inter-Plant Competition. Biometrics
**1967**, 23, 189–205. [Google Scholar] [CrossRef] [PubMed] - Moran, P.A.P. Notes on Continuous Stochastic Phenomena. Biometrika
**1950**, 37, 17–23. [Google Scholar] [CrossRef] [PubMed] - Lesage, J.; Kelly Pace, R. Introduction to Spatial Econometrics; CRC Press: Boca Raton, FL, USA, 2009; Volume 1. [Google Scholar]
- Cliff, A.D.; Ord, J.K. Spatial Processes: Models & Applications; Pion: London, UK, 1981. [Google Scholar]
- Griffith, D.A.; Layne, L.J. A Casebook for Spatial Statistical Data Analysis: A Compilation of Analyses of Different Thematic Data Sets; Oxford University Press: Oxford, UK, 1999. [Google Scholar]
- Ver Hoef, J.M.; Hanks, E.M.; Hooten, M.B. On the relationship between conditional (CAR) and simultaneous (SAR) autoregressive models. Spat. Stat.
**2018**, 25, 68–85. [Google Scholar] [CrossRef][Green Version] - Cliff, A.D.; Ord, J.K. Spatial Autocorrelation; Pion Limited: London, UK, 1973. [Google Scholar]
- Sokal, R.R.; Oden, N.L. Spatial autocorrelation in biology: 1. Methodology. Biol. J. Linn. Soc.
**1978**, 10, 199–228. [Google Scholar] [CrossRef][Green Version] - Sokal, R.R.; Oden, N.L. Spatial autocorrelation in biology: 2. Some biological implications and four applications of evolutionary and ecological interest. Biol. J. Linn. Soc.
**1978**, 10, 229–249. [Google Scholar] [CrossRef][Green Version] - Legendre, P. Spatial Autocorrelation: Trouble or New Paradigm? Ecology
**1993**, 74, 1659–1673. [Google Scholar] [CrossRef][Green Version] - Auchincloss, A.H.; Gebreab, S.Y.; Mair, C.; Diez Roux, A.V. A Review of Spatial Methods in Epidemiology, 2000–2010. Annu. Rev. Public Health
**2012**, 33, 107–122. [Google Scholar] [CrossRef][Green Version] - Anselin, L.; Florax, R.; Rey, S.J. Advances in Spatial Econometrics: Methodology, Tools and Applications; Springer: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Mustafa, A.; Van Rompaey, A.; Cools, M.; Saadi, I.; Teller, J. Addressing the determinants of built-up expansion and densification processes at the regional scale. Urban Stud.
**2018**, 55, 3279–3298. [Google Scholar] [CrossRef][Green Version] - Goodchild, M.F. Geographical information science. Int. J. Geogr. Inf. Syst.
**1992**, 6, 31–45. [Google Scholar] [CrossRef] - Bartlett, M.S. The Statistical Analysis of Spatial Pattern; Chapman and Hall: Boca Raton, FL, USA, 1975. [Google Scholar]
- Whittle, P. On stationary processes in the plane. Biometrika
**1954**, 41, 434–449. [Google Scholar] [CrossRef] - Ord, K. Estimation Methods for Models of Spatial Interaction. J. Am. Stat. Assoc.
**1975**, 70, 120–126. [Google Scholar] [CrossRef] - Cressie, N.A.C. Statistics for Spatial Data; Wiley: Hoboken, NJ, USA, 1993. [Google Scholar]
- Griffith, D.A. Estimating Spatial Autoregressive Model Parameters with Commercial Statistical Packages. Geogr. Anal.
**1988**, 20, 176–186. [Google Scholar] [CrossRef] - Griffith, D.A. Simplifying the normalizing factor in spatial autoregressions for irregular lattices. Pap. Reg. Sci.
**1992**, 71, 71–86. [Google Scholar] [CrossRef] - Griffith, D.A. Approximation of Gaussian spatial autoregressive models for massive regular square tessellation data. Int. J. Geogr. Inf. Sci.
**2015**, 29, 2143–2173. [Google Scholar] [CrossRef] - Griffith, D.A. Faster maximum likelihood estimation of very large spatial autoregressive models: an extension of the Smirnov–Anselin result. J. Stat. Comput. Simul.
**2004**, 74, 855–866. [Google Scholar] [CrossRef] - Griffith, D.A. Extreme eigenfunctions of adjacency matrices for planar graphs employed in spatial analyses. Linear Algebra Its Appl.
**2004**, 388, 201–219. [Google Scholar] [CrossRef] - Griffith, D.A. Eigenfunction properties and approximations of selected incidence matrices employed in spatial analyses. Linear Algebra Its Appl.
**2000**, 321, 95–112. [Google Scholar] [CrossRef] - Kelejian, H.H.; Prucha, I.R. A Generalized Moments Estimator for the Autoregressive Parameter in a Spatial Model. Int. Econ. Rev.
**1999**, 40, 509–533. [Google Scholar] [CrossRef][Green Version] - Kelejian, H.H.; Prucha, I.R. Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. J. Econom.
**2010**, 157, 53–67. [Google Scholar] [CrossRef][Green Version] - Walde, J.; Larch, M.; Tappeiner, G. Performance contest between MLE and GMM for huge spatial autoregressive models. J. Stat. Comput. Simul.
**2008**, 78, 151–166. [Google Scholar] [CrossRef] - Luo, Q.; Griffith, D.A.; Wu, H. The Moran Coefficient and the Geary Ratio: Some Mathematical and Numerical Comparisons. In Advances in Geocomputation; Springer International Publishing: Cham, Switzerland, 2017; pp. 253–269. [Google Scholar]
- Sahr, K.; White, D.; Kimerling, A.J. Geodesic Discrete Global Grid Systems. Cartogr. Geogr. Inf. Sci.
**2003**, 30, 121–134. [Google Scholar] [CrossRef] - Chun, Y.; Griffith, D.A. Spatial Statistics and Geostatistics: Theory and Applications for Geographic Information Science and Technology; SAGE Publications: Thousand Oaks, CA, USA, 2013. [Google Scholar]
- Griffith, D.A. Spatial Autocorrelation and Spatial Filtering: Gaining Understanding Through Theory and Scientific Visualization; Springer: Berlin, Germany, 2003. [Google Scholar]
- Jong, P.; Sprenger, C.; Veen, F. On Extreme Values of Moran’s I and Geary’s c. Geogr. Anal.
**1984**, 16, 17–24. [Google Scholar] [CrossRef] - Griffith, A.D.; Chun, Y. Spatial Autocorrelation and Uncertainty Associated with Remotely-Sensed Data. Remote Sens.
**2016**, 8, 535. [Google Scholar] [CrossRef] - Goslee, S.C. Behavior of Vegetation Sampling Methods in the Presence of Spatial Autocorrelation. Plant Ecol.
**2006**, 187, 203–212. [Google Scholar] [CrossRef]

1 | Ord, 1975, Appendix B, contains a typographical error: a minus should be inserted after the second equality sign of $\alpha $. |

2 | There is a typo in [36] (p. 864), the brace needs to be removed. |

**Figure 2.**Three regular surface partitionings: (

**a**) square configuration with rook adjacency; (

**b**) square configuration with queen adjacency; (

**c**) hexagonal configuration.

**Figure 3.**Fitted curves of the MC against $\rho $ for 70-by-70 ($n=4900$ ) surface partitionings: (

**a**) square rook case; (

**b**) square queen case; and, (

**c**) hexagonal case.

**Figure 4.**Fitted curves of the asymptotic variance versus $\widehat{\rho}$ for 150-by-150 ($n=22,500$ ) adjacencies: (

**a**) square rook case; (

**b**) square queen case; and, (

**c**) hexagonal case.

**Figure 5.**Variance comparison plots for 10-by-10 ($n=100$) tessellations: (

**a**) square rook case; (

**b**) square queen case; (

**c**) hexagonal case.

**Figure 6.**Convergence and variance ratio plots at $\rho =0$. (

**a1**,

**a2**) Square rook variance convergence and ratio plots; (

**b1**,

**b2**) square queen variance convergence and ratio plots; (

**c1**,

**c2**) hexagonal variance convergence and ratio plots.

**Figure 7.**Landscapes with different degrees of spatial autocorrelation (SA): (

**a**) a 100-by-100 lattice across which pseudo-random numbers from a standard normal distribution are randomly distributed; (

**b**) 184 census blocks of Wuhan (2010) with 0–19% (ranging from 7.83 to 39.74, with green representing low rates and red representing high rates) displayed; (

**c**) 100-by-100 pixels clipped from the Yellow Mountain region image (2002, with the green representing vegetation and buff-to-white strips depicting mountain ridges).

Geographical Configurations | Function Forms | Parameter Estimates |
---|---|---|

Square rook | $\widehat{\rho}=\frac{a}{1+{\mathit{e}}^{b\ast MC+c}}+\frac{d}{{\lambda}_{min}}$ | $\widehat{a}=2,\widehat{b}=-8,\widehat{c}=0,\widehat{d}=1$ |

Square queen | $\widehat{\rho}=\frac{a}{1+{\mathit{e}}^{{\left|MC+b\right|}^{c}}}+\frac{d}{{\lambda}_{min}}$ | $\widehat{a}=5.8,\widehat{b}=-0.96,\widehat{c}=8,\widehat{d}=1$ |

Hexagonal | $\widehat{a}=5.5,\widehat{b}=-0.96,\widehat{c}=6.7,\widehat{d}=1$ |

Landscape | $\mathbf{Asymptotic}\mathbf{Variance}\mathbf{for}\mathit{\rho}=0.$ |
---|---|

Square rook | $72/\left(36PQ+23P+23Q+36\right)$ |

Square queen | $2400/\left(600PQ+639P+639Q+794\right)$ |

Hexagonal | $180/\left(60PQ+55P+60Q+71\right)$ |

Geographical Configurations | Function Forms | $\mathbf{Parameter}\mathbf{Estimates}\left({\mathit{R}}^{2}\right)$ |
---|---|---|

Square rook | $Var{\left(\widehat{\rho}\right)}_{asy}=a\ast {a}_{0}\ast {G}^{b-1}$ $\ast {\left(1-G\right)}^{c-1}$ | $\widehat{a}=\frac{17.9181}{{n}^{0.5395}}+6.7144,\left(0.9999\right)$ $\widehat{b}=2.4,\widehat{c}=2.4$ |

Square queen | $\widehat{a}=\frac{32.4112}{{n}^{0.4482}}+6.4694,\left(0.9990\right)$ $\widehat{b}=\frac{2.0836}{{n}^{0.3519}}+2.1107,\left(0.9975\right)$ $\widehat{c}=\frac{1.0641}{{n}^{0.2982}}+2.2843,\left(0.9974\right)$ | |

Hexagonal | $Var{\left(\widehat{\rho}\right)}_{asy}=a\ast {a}_{0}\ast {G}^{c\ast G+b-1}\ast {\left(1-G\right)}^{e\ast G+d-1}$ | $\widehat{a}=\frac{30.1752}{{n}^{0.3065}}-0.0160,\left(0.9941\right)$ $\widehat{b}=\frac{3.0023}{{n}^{0.1121}}+0.5299,\left(0.9966\right)$ $\widehat{c}=\{\begin{array}{c}0.0110\ast {(\mathrm{ln}\left(n\right)-6)}^{2}+0.02357,n\le 400\\ -0.0233\ast {(\mathrm{ln}\left(n\right)-6)}^{2}+0.02357,n400\end{array},$ $\left(0.9917\right)$ $\widehat{d}={n}^{-0.0647\ast \mathrm{ln}\left(n\right)+0.4876}-0.0634,\left(0.9900\right)$ $\widehat{e}=\frac{\mathrm{ln}n}{3.35013}-1.1161,\left(0.9890\right)$ |

Treatment | ExactJacob | AppJacob1 | AppJacob2 | AppJacob3 |
---|---|---|---|---|

Original MLEs | SR, SQ, H | - | - | SQ, H |

Griffith’s MLEs (2015) | - | SR | SR | SQ, H |

Dataset | Random Landscape | Wuhan Census Blocks | Yellow Mountain Image |
---|---|---|---|

Response variable | Pseudo-random numbers | $3.67-11.79\ast {\mathit{e}}^{-6.59\ast YR}$ | $1.46\ast {\mathit{e}}^{4.78\ast NDVI}-2.70$ |

Null hypothesis (${H}_{0}$) | ${\rho}_{0}=0$ | ${\rho}_{0}=0.7$ | ${\rho}_{0}=0.95$ |

MC | 0.0051 | 0.5304 | 0.9077 |

$\widehat{\rho}$ | 0.0101 | 0.7080 | 0.9697 |

$sd\left(\widehat{\rho}\right)$ | 0.0139 | 0.0598 | 0.0019 |

z-score | 0.7254 | 0.1338 | 10.1762 |

p-value | 0.4682 | 0.8935 | <2.5314e-24 |

*95% CI of $\widehat{\rho}$ | [−0.0171, 0.0373] | [0.5908, 0.8252] | [0.9659, 0.9735] |

Conclusion | Failed to reject ${H}_{0}$ | Failed to reject ${H}_{0}$ | Reject ${H}_{0}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Luo, Q.; Griffith, D.A.; Wu, H. On the Statistical Distribution of the Nonzero Spatial Autocorrelation Parameter in a Simultaneous Autoregressive Model. *ISPRS Int. J. Geo-Inf.* **2018**, *7*, 476.
https://doi.org/10.3390/ijgi7120476

**AMA Style**

Luo Q, Griffith DA, Wu H. On the Statistical Distribution of the Nonzero Spatial Autocorrelation Parameter in a Simultaneous Autoregressive Model. *ISPRS International Journal of Geo-Information*. 2018; 7(12):476.
https://doi.org/10.3390/ijgi7120476

**Chicago/Turabian Style**

Luo, Qing, Daniel A. Griffith, and Huayi Wu. 2018. "On the Statistical Distribution of the Nonzero Spatial Autocorrelation Parameter in a Simultaneous Autoregressive Model" *ISPRS International Journal of Geo-Information* 7, no. 12: 476.
https://doi.org/10.3390/ijgi7120476