# Lagrange Multiplier Test for Spatial Autoregressive Model with Latent Variables

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Spatial Autoregressive Model with Latent Variable (SAR-LVs Model)

**W**

_{ij}is defined as 1 for the entity where the common side or the common vertex meets the region of concern, and

**W**

_{ij}is defined as 0 for other regions [20].

#### 2.2. Estimation of Score of Latent Variable

#### 2.3. The Error Distribution of The SAR-LVs Model

#### 2.4. Test of Dependency Spatial

#### 2.5. Estimation of Parameter of SAR-LVs Model

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Matrix Variate Normal Distribution

#### Appendix A.2. Properties of Matrix and Derivative of Matrix/Vector

#### Appendix A.3. Derivative of the Element of the Information Matrix for the SAR-LVs Model

- The first partial derivative of the log-likelihood function $L\left(\lambda ,\mathsf{\beta},\Theta ;\mathit{l}\right)$ to $\lambda $ based on the error distribution in Equation (12) where $\Theta =A{p}^{-1}{A}^{\prime}={p}^{-1}A{A}^{\prime};p=\left({\Lambda}_{y}^{\prime}{\Theta}_{\epsilon *}^{-1}{\Lambda}_{y}\right);A=\left(I-\lambda W\right)\mathrm{and}\mathsf{\epsilon}=A\mathit{l}-K\mathsf{\beta}$.
- The first partial derivative of $\Theta to\lambda $$$\frac{\partial \lambda}{\partial \Theta}={p}^{-1}\frac{\partial \left(\left(I-\lambda W\right){\left(I-\lambda W\right)}^{\prime}\right)}{\partial \lambda}=-{p}^{-1}\left(W{\left(I-\lambda W\right)}^{\prime}+\left(I-\lambda W\right){W}^{\prime}\right)$$$$\frac{\partial \lambda}{\partial \Theta}=-{p}^{-1}\left(W{A}^{\prime}+A{W}^{\prime}\right)$$
- The first partial derivative of $ln|\Theta |$ to $\lambda $Based on B.2$$\frac{\partial \mathrm{ln}\left|\Theta \right|}{\partial \lambda}=\mathrm{Tr}\left({\left({\mathrm{p}}^{-1}A{A}^{\prime}\right)}^{-1}\frac{\partial \Theta}{\partial \lambda}\right)=\mathrm{Tr}\left(\mathrm{p}\left({\mathrm{p}}^{-1}A{A}^{\prime}\right)\left(-{\mathrm{p}}^{-1}\left(W{A}^{\prime}+A{W}^{\prime}\right)\right)\right)=-2\mathrm{Tr}\left({A}^{-1}W\right)$$
- The first partial derivative of $In\left|A\right|$ to $\lambda $Based on B.2:$$\frac{\partial \mathrm{ln}\left|A\right|}{\partial \lambda}=\mathrm{Tr}\left({\left(I-\lambda W\right)}^{-1}\frac{\partial \left(I-\lambda W\right)}{\partial \lambda}\right)=\mathrm{Tr}\left({A}^{-1}W\right)$$
- The first partial derivative of ${A}^{-1}$ to $\lambda $Based on B.3:$$\frac{\partial \mathrm{ln}{A}^{-1}}{\partial \lambda}=-{\left(I-\lambda W\right)}^{-1}\left(-W\right){\left(I-\lambda W\right)}^{-1}={A}^{-1}W{A}^{-1}$$
- The first partial derivative of ${\mathsf{\epsilon}}^{\prime}{\Theta}^{-1}\mathsf{\epsilon}$ to $\lambda $$${\mathsf{\epsilon}}^{\prime}{\Theta}^{-1}\mathsf{\epsilon}=[{\left(Al-K\mathsf{\beta}\right)}^{\prime}{\left({\mathrm{p}}^{-1}A{A}^{\prime}\right)}^{-1}\left(Al-K\mathsf{\beta}\right)or{\mathsf{\epsilon}}^{\prime}{\Theta}^{-1}\mathsf{\epsilon}=\mathrm{p}{\left(l-{A}^{-1}K\mathsf{\beta}\right)}^{\prime}\left(l-{A}^{-1}K\mathsf{\beta}\right)$$Based on B.4:$$\frac{\partial \left({\mathsf{\epsilon}}^{\prime}{\Theta}^{-1}\mathsf{\epsilon}\right)}{\partial \lambda}=-2p{\left({A}^{-1}W{A}^{-1}K\mathsf{\beta}\right)}^{\prime}\left(l-{A}^{-1}K\mathsf{\beta}\right)$$The first partial derivative of the log-likelihood function $L\left(\lambda ,\mathsf{\beta},\Theta ;\mathit{l}\right)$ to $\lambda $ based on the error distribution in Equation (12), point b, c, and d$$\frac{\partial L\left(\lambda ,\mathsf{\beta},\Theta ;\mathit{l}\right)}{\partial \lambda}=-\frac{1}{2}\left(-2\mathrm{Tr}\left({A}^{-1}W\right)\right)-\mathrm{Tr}\left({A}^{-1}W\right)+p{\left({A}^{-1}W{A}^{-1}K\mathsf{\beta}\right)}^{\prime}\left(\mathit{l}-{A}^{-1}K\mathsf{\beta}\right)$$$$\frac{\partial L\left(\lambda ,\mathsf{\beta},\Theta ;\mathit{l}\right)}{\partial \lambda}=p{\left({A}^{-1}W{A}^{-1}K\mathsf{\beta}\right)}^{\prime}\left(\mathit{l}-{A}^{-1}K\mathsf{\beta}\right)$$

- The first partial derivative of the log-likelihood function $L\left(\lambda ,\mathsf{\beta},\Theta ;\mathit{l}\right)$ to $\mathsf{\beta}$ based on the error distribution in Equation (12)Based on B.5:$$\frac{\partial L\left(\lambda ,\mathsf{\beta},\Theta ;\mathit{l}\right)}{\partial \mathsf{\beta}}=p{\left({A}^{-1}K\right)}^{\prime}\left(\mathit{l}-{A}^{-1}K\mathsf{\beta}\right)$$
- The second partial derivative $\frac{{\partial}^{2}\mathit{l}\left(\lambda ,\mathsf{\beta},\Theta ;\mathit{l}\right)}{\partial {\lambda}^{2}}$Based on B.3:$$\frac{{\partial}^{2}L\left(\lambda ,\mathsf{\beta},\Theta ;\mathit{l}\right)}{\partial {\lambda}^{2}}=p[2{\left(K\mathsf{\beta}\right)}^{\prime}{\left(\left({A}^{-1}W{A}^{-1}\right){A}^{-1}\left({A}^{-1}W{A}^{-1}\right)\right)}^{\prime}\left(\mathit{l}-{A}^{-1}K\mathsf{\beta}\right)-$$$${\left({A}^{-1}W{A}^{-1}K\mathsf{\beta}\right)}^{\prime}\left({A}^{-1}W{A}^{-1}\right)\left(K\mathsf{\beta}\right)]$$
- The second partial derivative $\frac{{\partial}^{2}L\left(\lambda ,\mathsf{\beta},\Theta ;\mathit{l}\right)}{\partial \mathsf{\beta}\partial {\mathsf{\beta}}^{\prime}}$$$\frac{{\partial}^{2}L\left(\lambda ,\mathsf{\beta},\Theta ;\mathit{l}\right)}{\partial \mathsf{\beta}\partial {\mathsf{\beta}}^{\prime}}=-p{\left({A}^{-1}K\right)}^{\prime}\left({A}^{-1}K\right)$$
- The second partial derivative $\frac{{\partial}^{2}L\left(\lambda ,\mathsf{\beta},\Theta ;\mathit{l}\right)}{\partial \mathsf{\beta}\partial \lambda}$$$\frac{{\partial}^{2}L\left(\lambda ,\mathsf{\beta},\Theta ;\mathit{l}\right)}{\partial \mathsf{\beta}\partial \lambda}=p\left[{\left(\mathit{l}-{A}^{-1}K\mathsf{\beta}\right)}^{\prime}\left({A}^{-1}W{A}^{-1}K\right)-{\left({A}^{-1}W{A}^{-1}K\mathsf{\beta}\right)}^{\prime}\left({A}^{-1}K\right)\right]$$
- The element of the information matrix
- The Element (1,1), namely ${\stackrel{~}{\mathsf{\Psi}}}_{\lambda \lambda}$$${\stackrel{~}{\mathsf{\Psi}}}_{\lambda \lambda}=\mathrm{E}\left(-\frac{{\partial}^{2}L\left(\lambda ,\mathsf{\beta},\mathbf{\Theta};\mathit{l}\right)}{\partial {\lambda}^{2}}\right)$$$${\stackrel{~}{\mathsf{\Psi}}}_{\lambda \lambda}=p[{\left({\mathbf{A}}^{-1}\mathbf{W}{\mathbf{A}}^{-1}\mathbf{K}\mathsf{\beta}\right)}^{\prime}\left({\mathbf{A}}^{-1}\mathbf{W}{\mathbf{A}}^{-1}\right)\left(\mathbf{K}\mathsf{\beta}\right)-2{\left(\mathbf{K}\mathsf{\beta}\right)}^{\prime}{\left(\left({\mathbf{A}}^{-1}\mathbf{W}{\mathbf{A}}^{-1}\right){\mathbf{A}}^{-1}\left({\mathbf{A}}^{-1}\mathbf{W}{\mathbf{A}}^{-1}\right)\right)}^{\prime}$$$$\left(e{\mathsf{\eta}}_{t}-{\mathbf{A}}^{-1}\mathbf{K}\mathsf{\beta}\right)]$$$${\stackrel{~}{\mathsf{\Psi}}}_{\lambda \lambda}=p\left[{\left(WK\mathsf{\beta}\right)}^{\prime}\left(WK\mathsf{\beta}\right)-2{\left(WK\mathsf{\beta}\right)}^{\prime}W\left(\mathbf{e}{\mathsf{\eta}}_{t}-{A}^{-1}K\mathsf{\beta}\right)\right]$$
- The Element (2,2), namely ${\stackrel{~}{\mathsf{\psi}}}_{\mathsf{\beta}\mathsf{\beta}}$$${\stackrel{~}{\mathsf{\psi}}}_{\mathsf{\beta}\mathsf{\beta}}=\mathrm{E}\left(-\frac{{\partial}^{2}L\left(\lambda ,\mathsf{\beta},\mathbf{\Theta};\mathit{l}\right)}{\partial \mathsf{\beta}\partial {\mathsf{\beta}}^{\prime}}\right)=\mathrm{E}\left(-\left(-p{\left({\mathbf{A}}^{-1}\mathbf{K}\right)}^{\prime}\left({\mathbf{A}}^{-1}\mathbf{K}\right)\right)\right)=p{\left({\mathbf{A}}^{-1}\mathbf{K}\right)}^{\prime}\left({\mathbf{A}}^{-1}\mathbf{K}\right)$$
- The Element (1,2), namely ${\stackrel{~}{\mathsf{\psi}}}_{\lambda \mathsf{\beta}}$$${\stackrel{~}{\mathsf{\psi}}}_{\lambda \mathsf{\beta}}=\mathrm{E}\left(-\frac{{\partial}^{2}\mathrm{L}\left(\lambda ,\mathsf{\beta},\mathbf{\Theta};\mathit{l}\right)}{\partial \lambda \partial {\mathsf{\beta}}^{\prime}}\right)$$$${\stackrel{~}{\mathsf{\psi}}}_{\lambda \mathsf{\beta}}=p\left[{\left({\mathbf{A}}^{-1}\mathbf{W}{\mathbf{A}}^{-1}\mathbf{K}\mathsf{\beta}\right)}^{\prime}\left({\mathbf{A}}^{-1}\mathbf{K}\right)-{\left(e{\mathsf{\eta}}_{t}-{\mathbf{A}}^{-1}\mathbf{K}\mathsf{\beta}\right)}^{\prime}\left({\mathbf{A}}^{-1}\mathbf{W}{\mathbf{A}}^{-1}\mathbf{K}\right)\right]$$
- The Element (2,1), Namely ${\stackrel{~}{\mathsf{\psi}}}_{\mathsf{\beta}\lambda}$$${\stackrel{~}{\mathsf{\psi}}}_{\mathsf{\beta}\lambda}=\mathrm{E}\left(-\frac{{\partial}^{2}\mathrm{L}\left(\lambda ,\mathsf{\beta},\mathbf{\Theta};\mathit{l}\right)}{\partial \mathsf{\beta}\partial \lambda}\right)$$$${\stackrel{~}{\mathsf{\psi}}}_{\mathsf{\beta}\lambda}=p\left[{\left({\mathbf{A}}^{-1}\mathbf{K}\right)}^{\prime}\left({\mathbf{A}}^{-1}\mathbf{W}{\mathbf{A}}^{-1}\mathbf{K}\mathsf{\beta}\right)-{\left({\mathbf{A}}^{-1}\mathbf{W}{\mathbf{A}}^{-1}\mathbf{K}\right)}^{\prime}\left(e{\mathsf{\eta}}_{t}-{\mathbf{A}}^{-1}\mathbf{K}\mathsf{\beta}\right)\right]$$$${\stackrel{~}{\mathsf{\psi}}}_{\mathsf{\beta}\lambda}=p\left[{K}^{\prime}\left(WK\mathsf{\beta}\right)-{\left(WK\right)}^{\prime}\left(\mathbf{e}{\mathsf{\eta}}_{\mathbf{t}}-K\mathsf{\beta}\right)\right]$$

- The information matrixif $\lambda =0$ then the information matrix is ${\stackrel{~}{\mathsf{\Psi}}}_{\theta}=\left[\begin{array}{cc}\begin{array}{c}{\stackrel{~}{\mathsf{\Psi}}}_{\lambda \lambda}\\ {\stackrel{~}{\mathsf{\Psi}}}_{\mathsf{\beta}\lambda}\end{array}& \begin{array}{c}{\stackrel{~}{\mathsf{\Psi}}}_{\lambda \mathsf{\beta}}\\ {\stackrel{~}{\mathsf{\Psi}}}_{\mathsf{\beta}\mathsf{\beta}}\end{array}\end{array}\right]$
- Invers of the information matrix when $\lambda =0$If the partition matrix is $C=\left[\begin{array}{c}\begin{array}{cc}{C}_{1}& {C}_{2}\end{array}\\ \begin{array}{cc}{C}_{3}& {C}_{4}\end{array}\end{array}\right]$ then the invers matrix is ${C}^{-1}=\left[\begin{array}{c}\begin{array}{cc}{C}_{E1}& {C}_{E2}\end{array}\\ \begin{array}{cc}{C}_{E3}& {C}_{E4}\end{array}\end{array}\right]$ where${\mathbf{C}}_{E1}={\left({C}_{1}-{C}_{2}{C}_{4}^{-1}{C}_{3}\right)}^{-1}$; ${C}_{E2}=\left(-{C}_{E1}{C}_{2}{C}_{4}^{-1}\right)$; ${C}_{E3}=-{C}_{4}^{-1}{C}_{3}{C}_{E1}$; and ${C}_{E4}=\left({C}_{4}^{-1}-{C}_{4}^{-1}{C}_{3}{C}_{E2}\right)$The element (1,1) of the information matrix invers is ${\stackrel{~}{\mathsf{\Psi}}}_{\lambda \lambda}^{-1}={\left({\stackrel{~}{\mathsf{\Psi}}}_{\lambda \lambda}-{\stackrel{~}{\mathsf{\Psi}}}_{\lambda \beta}{({\stackrel{~}{\mathsf{\Psi}}}_{\beta \beta})}^{-1}{\stackrel{~}{\mathsf{\Psi}}}_{\beta \lambda}\right)}^{-1}$$${\stackrel{~}{\mathsf{\Psi}}}_{\lambda \lambda}^{-1}=(p{\left(WK\mathsf{\beta}\right)}^{\prime}\left(WK\mathsf{\beta}\right)-2{\left(WK\mathsf{\beta}\right)}^{\prime}{W}^{\prime}\left(\mathbf{e}{\mathsf{\eta}}_{\mathbf{t}}-K\mathsf{\beta}\right)-p({\left(WK\mathsf{\beta}\right)}^{\prime}\left(WK\mathsf{\beta}\right)-{\left(\mathbf{e}{\mathsf{\eta}}_{\mathbf{t}}-K\mathsf{\beta}\right)}^{\prime}$$$$W\left(WK\mathsf{\beta}\right)-\left(WK\mathsf{\beta}\right){W}^{\prime}\left(\mathbf{e}{\mathsf{\eta}}_{\mathbf{t}}-K\mathsf{\beta}\right)+{\left(\mathbf{e}{\mathsf{\eta}}_{\mathbf{t}}-K\mathsf{\beta}\right)}^{\prime}W{W}^{\prime}{\left(\mathbf{e}{\mathsf{\eta}}_{\mathbf{t}}-K\mathsf{\beta}\right)))}^{-1}$$$${\stackrel{~}{\mathsf{\Psi}}}_{\lambda \lambda}^{-1}={p}^{-1}{\left(-{\left(\mathbf{e}{\mathsf{\eta}}_{\mathbf{t}}-K\mathsf{\beta}\right)}^{\prime}W{W}^{\prime}\left(\mathbf{e}{\mathsf{\eta}}_{\mathbf{t}}-K\mathsf{\beta}\right)\right)}^{-1}$$$${\stackrel{~}{\mathsf{\Psi}}}_{\lambda \lambda}^{-1}=-{p}^{-1}{\left({\left(\mathbf{e}{\mathsf{\eta}}_{\mathbf{t}}-K\mathsf{\beta}\right)}^{\prime}W{W}^{\prime}\left(\mathbf{e}{\mathsf{\eta}}_{\mathbf{t}}-K\mathsf{\beta}\right)\right)}^{-1}$$

## References

- Bollen, K.A. Structural Equations with Latent Variables; John Wiley & Sons: New York, NY, USA, 1989. [Google Scholar]
- Civelek, M.E. Essentials of Structural Equation Modeling; The University of Nebraska: Lincoln, NE, USA, 2018; ISBN 978-1-60962-129-2. [Google Scholar]
- Wang, F.; Wall, M.M. Generalized common spatial factor model. Biostatistics
**2003**, 4, 569–582. [Google Scholar] [CrossRef] [PubMed] - Christensen, W.F.; Amemiya, Y. Latent variable analysis of multivariate spatial data. J. Am. Stat. Assoc.
**2002**, 97, 302–317. [Google Scholar] [CrossRef] - Hogan, J.W.; Tchernis, R. Bayesian Factor Analysis for Spatially Correlated Data, With Application to Summarizing Area-Level Material Deprivation from Census Data. J. Am. Stat. Assoc.
**2004**, 99, 314–324. [Google Scholar] [CrossRef] - Liu, X.; Wall, M.M.; Hodges, J.S. Generalized spatial structural equation models. Biostatistics
**2005**, 6, 539–557. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Congdon, P. A spatial structural equation model for health outcomes. J. Stat. Plan. Inference
**2008**, 138, 2090–2105. [Google Scholar] [CrossRef] - Oud, J.H.L.; Folmer, H. A Structural Equation Approach to Models with Spatial Dependence. Geogr. Anal.
**2008**, 40, 152–166. [Google Scholar] [CrossRef] - Anselin, L. Spatial Econometrics: Methods and Models; Kluwer Academic Publisher: Dordrecht, The Netherlands, 1988; Volume 4, ISBN 978-90-481-8311-1. [Google Scholar]
- Joreskog, K.G.; Sorbom, D. Lisrel 8: User’s Reference Guide; Scentific Software: Chicago, IL, USA, 1997. [Google Scholar]
- Anekawati, A.; Widjanarko Otok, B. Modelling of the education quality of a high schools in Sumenep Regency using spatial structural equation modelling. J. Phys. Conf. Ser.
**2017**, 890, 012094. [Google Scholar] [CrossRef] - Trujillo, G.S. Pathmox Approach: Segmentation Trees in Partial Least Squares Path Modeling; Universitat Politecnica de Catalunya: Barcelona, Spain, 2009. [Google Scholar]
- Anekawati, A.; Otok, B.W.; Sutikno, P. Generalized method of moments approach to spatial structural equation modeling. FJMS
**2018**, 103, 1057–1076. [Google Scholar] [CrossRef] - Kelejian, H.H.; Prucha, I.R. A Generalized Spatial Two-Stage Least Squares Procedure for Estimating a Spatial Autoregressive Model with Autoregressive Disturbances. J. Real Estate Financ. Econ.
**1998**, 17, 99–121. [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] - Jeong, S.; Yoon, D. Examining Vulnerability Factors to Natural Disasters with a Spatial Autoregressive Model: The Case of South Korea. Sustainability
**2018**, 10, 1651. [Google Scholar] [CrossRef] [Green Version] - Breusch, T.S.; Pagan, A.R. The Lagrange Multiplier Test and its Applications to Model Specification in Econometrics. Rev. Econ. Stud.
**1980**, 47, 239–253. [Google Scholar] [CrossRef] - Anselin, L. Lagrange Multiplier Test Diagnostics for Spatial Dependence and Spatial Heterogeneity. Geogr. Anal.
**1988**, 20, 1–17. [Google Scholar] [CrossRef] - Yang, Z. LM tests of spatial dependence based on bootstrap critical values. J. Econom.
**2015**, 185, 33–59. [Google Scholar] [CrossRef] - LeSage, J.P. The Theory and Practice of Spatial Econometrics, 1st ed.; The University of Toledo: Toledo, OH, USA, 1999. [Google Scholar]
- Gupta, A.K.; Nagar, D.K. Matrix Variate Distributions; Monographs and Surveys in Pure and Applied Mathematics; Chapman and Hall/CRC: New York, NY, USA, 2000; ISBN 1-58488-046-5. [Google Scholar]
- Otok, B.W.; Standsyah, R.E.; Suharsono, A.; Purhadi. Development of Model Poverty in Java Using Meta-Analysis Structural Equation Modeling (MASEM). In Proceedings of the 2nd International Conference on Science, Mathematics, Environment, and Education, Surakarta, Indonesia, 26–28 July 2019; AIP Conference Proceedings Volume 2194. p. 020078. [Google Scholar] [CrossRef]
- BPS. Sumenep in Figures 2018; BPS-Statistics of Sumenep Regency: Sumenep, Indonesia, 2018; ISBN 0215.2193.
- BPS. Data dan Informasi Kemiskinan Kabupaten/Kota tahun 2019; BPS-Statistics Indonesia: Jakarta, Indonesia, 2019.

**Table 1.**The estimation result of parameter and spatial autoregressive coefficient for the education model.

Variable | Coefficient |
---|---|

School Infrastructure (b1) | 2.3121 |

Socioeconomic condition (b2) | 0.1286 |

Constant (b0) | 9.6604 |

Spatial Autoregressive Coefficient (λ) | −0.002 |

District | Number of Public Senior High Schools | Number of Junior High School Graduate Students | Number of New Senior High School Students |
---|---|---|---|

Kalianget | 1 | 488 | 744 |

Kota Sumenep | 3 | 1582 | 2041 |

Manding | 0 | 201 | 93 |

Batuputih | 0 | 226 | 143 |

Gapura | 1 | 485 | 622 |

Batang Batang | 0 | 552 | 360 |

Dungkek | 0 | 409 | 156 |

**Table 3.**The estimation result of parameter and spatial autoregressive coefficient for the poverty model.

Variable | Coefficient |
---|---|

Economy (b1) | 0.0742 |

Human Resource (b2) | −0.0722 |

Health (b3) | 0.0155 |

Constant (b0) | 7.0881 |

Spatial Autoregressive Coefficient (λ) | 0.2345 |

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## Share and Cite

**MDPI and ACS Style**

Anekawati, A.; Otok, B.W.; Purhadi, P.; Sutikno, S.
Lagrange Multiplier Test for Spatial Autoregressive Model with Latent Variables. *Symmetry* **2020**, *12*, 1375.
https://doi.org/10.3390/sym12081375

**AMA Style**

Anekawati A, Otok BW, Purhadi P, Sutikno S.
Lagrange Multiplier Test for Spatial Autoregressive Model with Latent Variables. *Symmetry*. 2020; 12(8):1375.
https://doi.org/10.3390/sym12081375

**Chicago/Turabian Style**

Anekawati, Anik, Bambang Widjanarko Otok, Purhadi Purhadi, and Sutikno Sutikno.
2020. "Lagrange Multiplier Test for Spatial Autoregressive Model with Latent Variables" *Symmetry* 12, no. 8: 1375.
https://doi.org/10.3390/sym12081375