# The Biggest Myth in Spatial Econometrics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- This problem of what the spatial lag actually represents is bound up with the problem of definition of the spatial weights matrix, which is assumed to be a nonstochastic matrix capturing our hypothesis about the nature of the spatial interactions we are modelling. The problem is that, unlike the simple notion of a time series lag, the spatial lag is a very fluid and complex entity open to multiple definitions within a single study. Critics of spatial econometrics almost always in our experience home in on the arbitrary nature of the weights matrix, asking “how is it defined and why is it precisely like that when it could easily have been like this, what does it mean, and are not the results obtained conditional on somewhat arbitrary decisions taken about its structure?”. Some future research on the robustness of outcomes to variations in assumptions about the weight matrix structure would be helpful in allaying such criticisms, although ideally carefully structured arguments coming from theory and leading precisely to the typical reduced form spatial econometric model, with a spatial lag and exogenous lags also, are the preferred option.

## 2. Measures of Similarity between Weight Matrices

#### 2.1. Correlation between ${W}_{a}u$ and ${W}_{b}u$ for Varying W

#### 2.2. Measures of Correlation ${\mathit{between}}$ Predictions ${\mathit{from}}$ Varying W

#### 2.3. Measures of Correlation for Effects Estimates Based on Varying W

#### 2.4. Applied Illustrations of Similarity Measures between Varying W

Neighbors | 1st Order | 10th Order | Effects | Predictions |
---|---|---|---|---|

m | corr(${W}_{a}u,{W}_{b}u$) | corr(${W}_{a}^{10}u,{W}_{b}^{10}u$) | Estimates | |

5 | $0.7157$ | $0.7642$ | $0.9331$ | $0.9817$ |

6 | $0.6980$ | $0.7930$ | $0.9392$ | $0.9829$ |

7 | $0.6860$ | $0.8287$ | $0.9478$ | $0.9836$ |

8 | $0.6748$ | $0.8491$ | $0.9533$ | $0.9848$ |

9 | $0.6596$ | $0.8676$ | $0.9568$ | $0.9848$ |

10 | $0.6469$ | $0.8787$ | $0.9590$ | $0.9851$ |

11 | $0.6267$ | $0.8825$ | $0.9598$ | $0.9851$ |

12 | $0.6133$ | $0.8801$ | $0.9586$ | $0.9839$ |

13 | $0.5862$ | $0.8689$ | $0.9563$ | $0.9827$ |

14 | $0.5702$ | $0.8653$ | $0.9542$ | $0.9816$ |

15 | $0.5566$ | $0.8593$ | $0.9523$ | $0.9798$ |

16 | $0.5415$ | $0.8486$ | $0.9498$ | $0.9786$ |

17 | $0.5172$ | $0.8377$ | $0.9469$ | $0.9775$ |

18 | $0.4993$ | $0.8246$ | $0.9443$ | $0.9763$ |

19 | $0.4810$ | $0.8114$ | $0.9419$ | $0.9751$ |

20 | $0.4688$ | $0.8012$ | $0.9399$ | $0.9738$ |

21 | $0.4623$ | $0.7854$ | $0.9375$ | $0.9721$ |

22 | $0.4460$ | $0.7772$ | $0.9350$ | $0.9709$ |

23 | $0.4405$ | $0.7706$ | $0.9337$ | $0.9703$ |

24 | $0.4343$ | $0.7643$ | $0.9320$ | $0.9696$ |

25 | $0.4208$ | $0.7551$ | $0.9300$ | $0.9687$ |

26 | $0.4066$ | $0.7464$ | $0.9281$ | $0.9681$ |

27 | $0.3913$ | $0.7368$ | $0.9263$ | $0.9671$ |

28 | $0.3854$ | $0.7255$ | $0.9244$ | $0.9663$ |

29 | $0.3760$ | $0.7135$ | $0.9224$ | $0.9651$ |

30 | $0.3735$ | $0.7056$ | $0.9209$ | $0.9644$ |

## 3. Origins of the Myth

#### 3.1. Past Literature

#### 3.2. A Re-Examination of Bell and Bockstael (2000)

- What emerges from the example is that our results are more sensitive to the specification of the spatial weight matrix than to estimation technique. Compared to the variation across estimation methods, the results across spatial weight matrices are much less stable.
- Where qualitative results change, they are almost universally associated with changes in the spatial weight matrix and not with changes in the estimation method. For three of the estimated coefficients, one spatial weight matrix produces results qualitatively different from the others, and, for three more of the estimated coefficients, two spatial weight matrices produce results qualitatively different from the other two. There is no particular pattern to these reversals, nor is there a pattern when comparing the spatial correlation-corrected results to the OLS results.

OLS | ML | t–Statistic (t–Probability) | |
---|---|---|---|

${\widehat{\beta}}_{o}$ (${\widehat{\sigma}}_{{\beta}_{o}}$) | ${\widehat{\beta}}_{\mathrm{ml}}$ (${\widehat{\sigma}}_{{\beta}_{\mathrm{ml}}}$) | ${H}_{o}:{\widehat{\beta}}_{o}={\widehat{\beta}}_{\mathrm{ml}}$ | |

Intercept | 4.7332 (0.2047) | 5.1725 (0.2204) | 1.9932 (0.0465) |

LIV | 0.6926 (0.0124) | 0.6537 (0.0135) | 2.8815 (0.0040) |

LLT | 0.0079 (0.0052) | 0.0002 (0.0052) | 1.4808 (0.1390) |

LDC | −0.1494 (0.0195) | −0.1774 (0.0245) | 1.1429 (0.2534) |

LBA | −0.0453 (0.0114) | −0.0169 (0.0156) | 1.8205 (0.0690) |

POPN | −0.0493 (0.0408) | −0.0149 (0.0414) | 0.8309 (0.4062) |

PNAT | 0.0799 (0.0177) | 0.0586 (0.0212) | 1.0047 (0.3153) |

PDEV | 0.0677 (0.0180) | 0.0253 (0.0253) | 1.6759 (0.0941) |

PLOW | −0.0166 (0.0194) | −0.0374 (0.0224) | 0.9286 (0.3533) |

PSEW | −0.1187 (0.0173) | −0.0828 (0.0180) | 1.9944 (0.0464) |

#### 3.3. A Re-Examination of Kostov (2010)

Variable | Description |
---|---|

CRIME | per capita crime rate by town |

CHARLES | Charles River dummy variable (=1 if tract bounds river; 0 otherwise) |

NOX | nitric oxides concentration (parts per 10 million) |

ROOMS | average number of rooms per dwelling |

DISTANCE | weighted distances to five Boston employment centers |

RADIAL | index of accessibility to radial highways |

TAX | full-value property-tax rate per $10,000 |

PTRATIO | pupil-teacher ratio by town |

B | 1000(Bk - 0.63)${}^{2}$ where Bk is the proportion of blacks by town |

LSTATUS | % lower status of the population |

γ | $m=5$ | $m=6$ | $m=7$ |
---|---|---|---|

0 | 0.0001 | 0.0095 | 0.0007 |

0.1 | 0.0004 | 0.0288 | 0.0025 |

0.2 | 0.0013 | 0.0726 | 0.0083 |

0.3 | 0.0028 | 0.1381 | 0.0207 |

0.4 | 0.0041 | 0.1835 | 0.0368 |

0.5 | 0.0041 | 0.1672 | 0.0445 |

0.6 | 0.0029 | 0.1080 | 0.0376 |

0.7 | 0.0015 | 0.0525 | 0.0234 |

0.8 | 0.0007 | 0.0203 | 0.0114 |

0.9 | 0.0002 | 0.0065 | 0.0045 |

1 | 0.0001 | 0.0018 | 0.0015 |

1.1 | 0.0000 | 0.0004 | 0.0004 |

1.2 | 0.0000 | 0.0001 | 0.0001 |

1.3 | 0.0000 | 0.0000 | 0.0000 |

1.4 | 0.0000 | 0.0000 | 0.0000 |

1.5 | 0.0000 | 0.0000 | 0.0000 |

1.6 | 0.0000 | 0.0000 | 0.0000 |

1.7 | 0.0000 | 0.0000 | 0.0000 |

1.8 | 0.0000 | 0.0000 | 0.0000 |

1.9 | 0.0000 | 0.0000 | 0.0000 |

2 | 0.0000 | 0.0000 | 0.0000 |

Direct Effects CRIME | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.0100 | −0.0079 | −0.0079 | −0.0058 |

0.4 | −0.0101 | −0.0079 | −0.0080 | −0.0058 |

0.6 | −0.0102 | −0.0080 | −0.0080 | −0.0059 |

0.8 | −0.0102 | −0.0081 | −0.0082 | −0.0060 |

1 | −0.0104 | −0.0082 | −0.0083 | −0.0061 |

Direct Effects CHARLES | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.0374 | 0.0189 | 0.0263 | 0.0753 |

0.4 | −0.0358 | 0.0198 | 0.0279 | 0.0754 |

0.6 | −0.0336 | 0.0228 | 0.0286 | 0.0792 |

0.8 | −0.0308 | 0.0263 | 0.0308 | 0.0835 |

1 | −0.0271 | 0.0299 | 0.0342 | 0.0869 |

Direct Effects NOX${}^{\mathbf{2}}$ | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.4767 | −0.2889 | −0.2879 | −0.1010 |

0.4 | −0.4908 | −0.2966 | −0.2927 | −0.1023 |

0.6 | −0.4951 | −0.3085 | −0.3147 | −0.1219 |

0.8 | −0.5136 | −0.3190 | −0.3252 | −0.1245 |

1 | −0.5184 | −0.3267 | −0.3365 | −0.1350 |

Direct Effects ROOMS${}^{\mathbf{2}}$ | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | 0.0051 | 0.0073 | 0.0074 | 0.0095 |

0.4 | 0.0052 | 0.0073 | 0.0074 | 0.0093 |

0.6 | 0.0052 | 0.0073 | 0.0074 | 0.0094 |

0.8 | 0.0052 | 0.0073 | 0.0074 | 0.0095 |

1 | 0.0053 | 0.0074 | 0.0074 | 0.0095 |

Direct Effects DISTANCE | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.1976 | −0.1521 | −0.1481 | −0.1066 |

0.4 | −0.1949 | −0.1506 | −0.1468 | −0.1062 |

0.6 | −0.1922 | −0.1486 | −0.1461 | −0.1050 |

0.8 | −0.1920 | −0.1468 | −0.1448 | −0.1016 |

1 | −0.1924 | −0.1466 | −0.1443 | −0.1007 |

Direct Effects RAD | ||||
---|---|---|---|---|

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | 0.0467 | 0.0766 | 0.0760 | 0.1065 |

0.4 | 0.0460 | 0.0769 | 0.0769 | 0.1078 |

0.6 | 0.0468 | 0.0776 | 0.0773 | 0.1083 |

0.8 | 0.0478 | 0.0783 | 0.0781 | 0.1088 |

1 | 0.0473 | 0.0791 | 0.0777 | 0.1109 |

Direct Effects TAX | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.0005 | −0.0003 | −0.0003 | −0.0001 |

0.4 | −0.0005 | −0.0003 | −0.0003 | −0.0001 |

0.6 | −0.0005 | −0.0003 | −0.0003 | −0.0001 |

0.8 | −0.0005 | −0.0003 | −0.0003 | −0.0001 |

1 | −0.0005 | −0.0003 | −0.0003 | −0.0002 |

Direct Effects PTRATIO | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.0203 | −0.0120 | −0.0117 | −0.0038 |

0.4 | −0.0209 | −0.0125 | −0.0123 | −0.0040 |

0.6 | −0.0215 | −0.0133 | −0.0130 | −0.0050 |

0.8 | −0.0223 | −0.0137 | −0.0139 | −0.0052 |

1 | −0.0229 | −0.0145 | −0.0148 | −0.0061 |

Direct Effects B | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | 0.0001 | 0.0003 | 0.0003 | 0.0005 |

0.4 | 0.0001 | 0.0003 | 0.0003 | 0.0005 |

0.6 | 0.0001 | 0.0003 | 0.0003 | 0.0005 |

0.8 | 0.0001 | 0.0003 | 0.0003 | 0.0005 |

1 | 0.0001 | 0.0003 | 0.0003 | 0.0005 |

Direct Effects LSTATUS | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.3059 | −0.2641 | −0.2619 | −0.2223 |

0.4 | −0.3019 | −0.2603 | −0.2592 | −0.2186 |

0.6 | −0.3015 | −0.2594 | −0.2578 | −0.2173 |

0.8 | −0.3000 | −0.2581 | −0.2571 | −0.2161 |

1 | −0.3002 | −0.2574 | −0.2584 | −0.2146 |

Indirect Effects CRIME | ||||
---|---|---|---|---|

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.0094 | −0.0073 | −0.0068 | −0.0052 |

0.4 | −0.0094 | −0.0073 | −0.0067 | −0.0052 |

0.6 | −0.0091 | −0.0069 | −0.0065 | −0.0048 |

0.8 | −0.0088 | −0.0067 | −0.0063 | −0.0046 |

1 | −0.0086 | −0.0064 | −0.0060 | −0.0042 |

Indirect Effects CHARLES | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.0393 | 0.0171 | 0.0226 | 0.0734 |

0.4 | −0.0375 | 0.0181 | 0.0239 | 0.0737 |

0.6 | −0.0369 | 0.0195 | 0.0230 | 0.0759 |

0.8 | −0.0359 | 0.0212 | 0.0233 | 0.0784 |

1 | −0.0341 | 0.0229 | 0.0245 | 0.0799 |

Indirect Effects NOX${}^{\mathbf{2}}$ | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.4553 | −0.2675 | −0.2481 | −0.0796 |

0.4 | −0.4652 | −0.2710 | −0.2481 | −0.0768 |

0.6 | −0.4537 | −0.2672 | −0.2532 | −0.0806 |

0.8 | −0.4575 | −0.2630 | −0.2456 | −0.0684 |

1 | −0.4462 | −0.2545 | −0.2428 | −0.0629 |

Indirect Effects ROOMS${}^{\mathbf{2}}$ | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | 0.0046 | 0.0067 | 0.0064 | 0.0087 |

0.4 | 0.0046 | 0.0067 | 0.0063 | 0.0088 |

0.6 | 0.0042 | 0.0063 | 0.0060 | 0.0085 |

0.8 | 0.0040 | 0.0061 | 0.0057 | 0.0082 |

1 | 0.0036 | 0.0057 | 0.0054 | 0.0079 |

Indirect Effects DISTANCE | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.1855 | −0.1406 | −0.1275 | −0.0957 |

0.4 | −0.1823 | −0.1366 | −0.1249 | −0.0909 |

0.6 | −0.1745 | −0.1290 | −0.1182 | −0.0834 |

0.8 | −0.1664 | −0.1209 | −0.1111 | −0.0754 |

1 | −0.1582 | −0.1130 | −0.1036 | −0.0679 |

Indirect Effects RAD | ||||
---|---|---|---|---|

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | 0.0401 | 0.0708 | 0.0651 | 0.1016 |

0.4 | 0.0395 | 0.0701 | 0.0650 | 0.1007 |

0.6 | 0.0366 | 0.0674 | 0.0621 | 0.0983 |

0.8 | 0.0318 | 0.0636 | 0.0597 | 0.0955 |

1 | 0.0298 | 0.0612 | 0.0567 | 0.0926 |

Indirect Effects TAX | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.0005 | −0.0003 | −0.0003 | −0.0001 |

0.4 | −0.0005 | −0.0003 | −0.0003 | −0.0001 |

0.6 | −0.0005 | −0.0003 | −0.0003 | −0.0001 |

0.8 | −0.0004 | −0.0003 | −0.0003 | −0.0001 |

1 | −0.0004 | −0.0003 | −0.0002 | −0.0001 |

Indirect Effects PTRATIO | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.0193 | −0.0111 | −0.0100 | −0.0029 |

0.4 | −0.0198 | −0.0113 | −0.0103 | −0.0029 |

0.6 | −0.0197 | −0.0115 | −0.0106 | −0.0033 |

0.8 | −0.0198 | −0.0112 | −0.0106 | −0.0027 |

1 | −0.0197 | −0.0113 | −0.0107 | −0.0029 |

Indirect Effects B | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | 0.0001 | 0.0003 | 0.0003 | 0.0004 |

0.4 | 0.0001 | 0.0003 | 0.0003 | 0.0004 |

0.6 | 0.0001 | 0.0002 | 0.0002 | 0.0004 |

0.8 | 0.0001 | 0.0002 | 0.0002 | 0.0004 |

1 | 0.0000 | 0.0002 | 0.0002 | 0.0004 |

Indirect Effects LSTATUS | ||||

Decay | −2$\mathbf{\sigma}$ | $\mathit{m}\mathbf{=}\mathbf{6}$ | $\mathit{m}\mathbf{=}\mathbf{5}$ | +2$\mathbf{\sigma}$ |

0.2 | −0.2876 | −0.2458 | −0.2283 | −0.2039 |

0.4 | −0.2804 | −0.2388 | −0.2215 | −0.1971 |

0.6 | −0.2668 | −0.2247 | −0.2097 | −0.1827 |

0.8 | −0.2545 | −0.2126 | −0.1976 | −0.1706 |

1 | −0.2416 | −0.1988 | −0.1870 | −0.1560 |

#### 3.4. A Diagnostic Example

Model | SAR Model | SDM Model |
---|---|---|

Posterior Probability | Posterior Probability | |

W-contiguity | 0.0000 | 0.0000 |

neighbors = 3 | 0.0000 | 0.0000 |

neighbors = 4 | 0.0000 | 0.0000 |

neighbors = 5 | 0.0000 | 0.0000 |

neighbors = 6 | 0.0000 | 0.0000 |

neighbors = 7 | 0.0000 | 0.0000 |

neighbors = 8 | 0.0000 | 0.0000 |

neighbors = 9 | 0.0000 | 0.0000 |

neighbors = 10 | 0.0000 | 0.0000 |

neighbors = 11 | 0.0000 | 0.0000 |

neighbors = 12 | 0.0000 | 0.0000 |

neighbors = 13 | 0.0000 | 0.0000 |

neighbors = 14 | 0.0093 | 0.0030 |

neighbors = 15 | 0.9905 | 0.9940 |

neighbors = 16 | 0.0003 | 0.0030 |

neighbors = 17 | 0.0000 | 0.0000 |

neighbors = 18 | 0.0000 | 0.0000 |

neighbors = 19 | 0.0000 | 0.0000 |

neighbors = 20 | 0.0000 | 0.0000 |

## 4. Interaction between Specification of the Model and $\mathit{W}$ Matrix

Neighbors | Total Impact | Likelihood | |
---|---|---|---|

4 | $1.547$ | $-1480.282$ | |

5 | $1.629$ | $-1436.294$ | |

6 | $1.685$ | $-1418.920$ | |

7 | $1.728$ | $-1404.318$ | |

8 | $1.754$ | $-1401.690$ | |

9 | $1.781$ | $-1393.235$ | |

10 | $1.807$ | $-1379.055$ | |

11 | $1.831$ | $-1369.690$ | |

12 | $1.853$ | $-1359.639$ | |

13 | $1.872$ | $-1352.425$ | |

14 | $1.890$ | $-1352.786$ | |

15 | $1.903$ | $-1356.933$ | |

16 | $1.916$ | $-1359.539$ | |

17 | $1.928$ | $-1358.922$ | |

18 | $1.940$ | $-1354.247$ | |

19 | $1.941$ | $-1360.217$ | |

20 | $1.951$ | $-1364.012$ | |

21 | $1.947$ | $-1366.057$ | |

22 | $1.946$ | $-1374.467$ | |

23 | $1.945$ | $-1376.323$ | |

24 | $1.946$ | $-1376.508$ | |

25 | $1.940$ | $-1379.946$ | |

26 | $1.940$ | $-1381.173$ | |

27 | $1.948$ | $-1381.586$ | |

28 | $1.955$ | $-1379.777$ | |

29 | $1.958$ | $-1383.759$ | |

30 | $1.961$ | $-1382.689$ |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- J.P. LeSage, and R.K. Pace. Introduction to Spatial Econometrics. Boca Raton, FL, USA: CRC Press. Boca Raton, FL, USA: Taylor & Francis, 2009. [Google Scholar]
- G. Arbia, and B. Fingleton. “New spatial econometric techniques and applications in Regional Science.” Pap. Reg. Sci. 87 (2008): 311–317. [Google Scholar] [CrossRef]
- R. Barry, and R.K. Pace. “Monte Carlo estimates of the log determinant of large sparse matrices.” Linear Algebr. Appl. 289 (1999): 41–54. [Google Scholar] [CrossRef]
- H.H. Kelejian, and I.R. Prucha. “Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances.” J. Econom. 157 (2010): 53–67. [Google Scholar] [CrossRef] [PubMed]
- L. Anselin. “Under the hood: Issues in the specification and interpretation of spatial regression models.” Agric. Econ. 27 (2002): 247–267. [Google Scholar] [CrossRef]
- P. Kostov. “Model boosting for spatial weighting matrix selection in spatial lag models.” Environ. Plann. B 37 (2010): 533–549. [Google Scholar] [CrossRef]
- K.P. Bell, and N.E. Bockstael. “Applying the Generalized-Moments Estimation Approach to Spatial Problems Involving Microlevel Data.” Rev. Econ. Statist. 87 (2000): 72–82. [Google Scholar] [CrossRef]
- R.K. Pace, J.P. LeSage, and S. Zhu. “Spatial Dependence in Regressors.” In Advances in Econometrics. Edited by D. Terrell and D. Millimet. Edited by D. Terrell and D. Millimet. Amsterdam, The Netherlands: Elsevier Science, 2012, Volume 30, pp. 257–295. [Google Scholar]
- H.H. Kelejian, and I.R. Prucha. “A Generalized Spatial Two-State Least Squares Procedure for estimating a Spatial Autogressive Model.” J. Real Estate Financ. Econ. 17 (1999): 99–121. [Google Scholar] [CrossRef]
- L. Anselin. Spatial Econometrics: Methods and Models. Dordrecht, The Netherlands: Kluwer Academic, 1988. [Google Scholar]
- R.K. Pace, and J.P. LeSage. “A Spatial Hausman Test.” Econ. Lett. 101 (2008): 282–284. [Google Scholar] [CrossRef]
- D. Harrison, and D.L. Rubinfeld. “Hedonic prices and the demand for clean air.” J. Environ. Econ. Manag. 5 (1978): 81–102. [Google Scholar] [CrossRef]
- O.W. Gilley, and R.K. Pace. “The Harrison and Rubinfeld Data Revisited.” J. Environ. Econ. Manag. 31 (1996): 403–405. [Google Scholar] [CrossRef]
- L.W. Hepple. “Bayesian model choice in spatial econometrics.” In Advances in Econometrics. Edited by J.P. LeSage and R.K. Pace. Oxford, UK: Elsevier Ltd., 2004, Volume 18, pp. 101–126. [Google Scholar]
- H.H. Kelejian, and G. Piras. “An extension of Kelejian’s J-test for non-nested spatial models.” Reg. Sci. Urban Econ. 41 (2011): 281–292. [Google Scholar] [CrossRef]
- L.M. Gerkman, and N. Ahlgren. “Practical Proposals for Specifying k-Nearest Neighbours Weights Matrices.” Spat. Econ. Anal. 9 (2014): 260–283. [Google Scholar] [CrossRef]
- J.P. LeSage, and R.K. Pace. “Interpreting Spatial Econometric Models.” In Handbook of Regional Science. Edited by M.M. Fischer and P. Nijkamp. Berlin, Germany: Springer, 2014, pp. 1535–1552. [Google Scholar]
- J.P. Elhorst. “Applied Spatial Econometrics: Raising the Bar.” Spat. Econ. Anal. 5 (2010): 9–28. [Google Scholar] [CrossRef]
- R.K. Pace, and S. Zhu. “Separable spatial modeling of spillovers and disturbances.” J. Geogr. Syst. 14 (2012): 75–90. [Google Scholar] [CrossRef]
- H.H. Kelejian, and P. Mukerji. “Important dynamic indices in spatial models.” Pap. Reg. Sci. 90 (2011): 693–702. [Google Scholar] [CrossRef]
- H.H. Kelejian, G.S. Tavlas, and G. Hondronyiannis. “A Spatial Modeling Approach to Contagion Among Emerging Economies.” Open Econ. Rev. 17 (2006): 423–442. [Google Scholar] [CrossRef]
- J.P. LeSage. “Spatial econometric panel data model specification: A Bayesian approach.” Spat. Stat. 9 (2014): 122–145. [Google Scholar] [CrossRef]
- S.-Y. Lee. “Three essays on spatial econometrics and empirical industrial organization.” Ph.D. Dissertation, The Ohio State University, Columbus, OH, USA, 2008; p. 131. [Google Scholar]
- R.K. Pace, and R.P. Barry. “Quick computation of spatial autoregressive estimators.” Geogr. Anal. 29 (1997): 232–246. [Google Scholar] [CrossRef]
- R.K. Pace, and J.P. LeSage. “Omitted variables biases of OLS and spatial lag models.” In Progress in Spatial Analysis: Theory and Computation, and Thematic Applications. Edited by A. Páez, J. Le Gallo, R. Buliung and S. Dall’Erba. Berlin, Germany: Springer, 2010, pp. 17–28. [Google Scholar]

^{1}Theoretical bounds for this parameter are set forth in LeSage and Pace [1], and depend on minimum and maximum eigenvalues of the spatial weight matrix W.^{2}In linear algebra a row-stochastic matrix has non-negative entries and each row sums to 1 while a doubly stochastic matrix is non-negative and both the rows and columns sum to 1. Symmetric, doubly stochastic weight matrices, although they have not been used as much in applications, have a number of simple theoretical properties (all real eigenvalues, eigenvectors, and are constant preserving). Sometimes, as in Arbia and Fingleton [2], weight matrices are said to be nonstochastic which means the elements are not random variables. So, from a statistical viewpoint a matrix could be termed nonstochastic while from a linear algebra view the same matrix could be said to be row- or doubly-stochastic. This is merely a difference in terminology that uses the term stochastic in different contexts. We make the traditional assumption that the matrix W is fixed in repeated sampling and therefore exogenously determined.^{3}We assume for simplicity that ${m}_{a}<{m}_{b}$, without loss of generality.^{5}If ${W}_{a}$ is much sparser than ${W}_{b}$, the maximum positive correlation between ${W}_{a}^{j}u$ and ${W}_{b}^{l}u$ might occur for powers $j>l$.^{6}It is not surprising there are few reports of cases where estimates were found to be sensitive to the choice used by the practitioner.^{7}We focus our discussion on their maximum likelihood estimates constructed using an inverse distance weight matrix based on a 600 m cut-off, but similar statements apply to maximum likelihood and GMM estimates based on the other three weight matrices.^{8}The standard errors from maximum likelihood SEM estimates were used to construct the $t-$test, since these are not adversely impacted by spatial dependence in the disturbances. However, this is not a multivariate test for the difference between the two vectors, but is a way of describing the differences between estimates.^{9}This result is based on assigning no prior distributions for the parameters $\beta ,{\sigma}^{2}$, and a uniform prior for ρ, over the interval D defined by the minimum and maximum eigenvalues of the matrix ${W}_{i}$ (Hepple 2004, p. 111).^{11}Decay values of $\gamma =0$ were also calculated, but to save space and make the tables smaller this value was excluded from the tables.^{12}Pace, LeSage and Zhu [8] suggest using median effects estimates since the total effects can have a non-symmetric distribution. In our case with 506 observations the means and medians produced nearly identical results.^{13}These can be produced using the lmarginal_cross_section function from the Spatial Econometrics Toolbox available at: www.spatial-econometrics.com.^{14}See Pace and Zhu [19] for a detailed discussion of this type of model specification.

© 2014 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 license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

LeSage, J.P.; Pace, R.K.
The Biggest Myth in Spatial Econometrics. *Econometrics* **2014**, *2*, 217-249.
https://doi.org/10.3390/econometrics2040217

**AMA Style**

LeSage JP, Pace RK.
The Biggest Myth in Spatial Econometrics. *Econometrics*. 2014; 2(4):217-249.
https://doi.org/10.3390/econometrics2040217

**Chicago/Turabian Style**

LeSage, James P., and R. Kelley Pace.
2014. "The Biggest Myth in Spatial Econometrics" *Econometrics* 2, no. 4: 217-249.
https://doi.org/10.3390/econometrics2040217