 




R. Kelley Pace 
LREC Endowed Chair of Real Estate 
Department of Finance, Louisiana State University 
Baton Rouge, LA 708036308 
OFF: (225)5786256, FAX: (225)5786095 
kelley@pace.am, www.spatialstatistics.com 
James P. LeSage 
Department of Economics, University of Toledo 
Toledo, OH 43606 
www.spatialeconometrics.com 
KEYWORDS: spatial statistics, spatial autoregression, nearest neighbor, maximum likelihood, sparse matrices, log determinant bounds, matrix determinant approximations, doubly stochastic, spatial data mining.

Definition 1. Let represent a sequence of individual nearest neighbor weight matrices such that if observation is the th nearest neighbor to observation and 0 otherwise . Let


Define the overall spatial weight matrix as a combination of the primitive spatial basis matrices:


where:


Finally, let represent a by matrix containing all possible traces of pairwise multiplications of the basis matrices. Specifically, for . 
(3) 
and since equals the first two terms of the series (given while for , since is nonnegative. Hence, and thus the upper bound is proved. 

For the lower bound, symmetry of implies real eigenvalues such that for . Even powers have all nonnegative eigenvalues since for positive integer . Recall the trace equals the sum of the real eigenvalues, and for positive integer , Since the maximum value for is 1 for any positive integer and since , . 

In particular, . Since . However, . Moreover, . Hence, and thus the lower bound is proved. QED. 
Since and meets the conditions of Proposition 1, the Corollary is proved. QED. 
(4) 
(5) 
(6) 
(7) 
(8) 
(9)  
(10) 
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Variables 
OLS 
Lower Bound 
Exact ML Lower 
Upper Bound 
ML 
Bound Weights 
ML  
CRIM 
0.0118 
0.0072 
0.0069 
0.0067 
ZN 
0.0001 
0.0004 
0.0004 
0.0004 
INDUS 
0.0002 
0.0009 
0.0010 
0.0011 
CHAS 
0.0921 
0.0146 
0.0114 
0.0077 
NOX 
0.6372 
0.2524 
0.2346 
0.2161 
RM 
0.0063 
0.0072 
0.0072 
0.0073 
AGE 
0.0001 
0.0004 
0.0005 
0.0005 
LDIS 
0.1978 
0.1643 
0.1616 
0.1599 
LRAD 
0.0896 
0.0592 
0.0574 
0.0560 
TAX 
0.0004 
0.0003 
0.0003 
0.0003 
PTRATIO 
0.0296 
0.0092 
0.0083 
0.0073 
B 
0.0004 
0.0003 
0.0003 
0.0003 
LSTAT 
37.4895 
23.3274 
22.4674 
21.7632 
0 
0.5150 
0.5400 
0.5650  
Log like 
700.35 
584.4019 
580.1425 
576.7777 
Variables 
OLS 
Restricted 
Lower 
Restricted 
Upper 
Restricted 
Log Like 
Bound 
Log Like 
Bound 
Log Like  
Land Area 
0.0025 
264,834 
0.0003 
227,572 
0.0003 
224,099 
Pop 
0.0253 
261,261 
0.0200 
227,373 
0.0189 
224,098 
Income 
0.6628 
282,537 
0.6471 
244,844 
0.6400 
239,264 
Age 
0.1338 
260,785 
0.1337 
228,754 
0.1326 
225,448 
0.8850 
260,176 
0.9850 
260,176  
93 
93 
93 

0 
23 
23 

Unrestricted 
260,176 
227,218 
223,995  
Log Like 
Variables 
OLS 
Restricted 
Lower 
Restricted 
Upper 
Restricted 
Log Like 
Bound 
Log Like 
Bound 
Log Like  
Land Area 
0.0850 
271,195 
0.0177 
232,756 
0.0082 
230,914 
Pop 
0.1146 
267,184 
0.0195 
232,526 
0.0142 
230,803 
Income 
1.0837 
288,183 
0.4840 
246,472 
0.4358 
240,730 
Age 
0.1269 
267,093 
0.0763 
233,199 
0.0700 
231,408 
0 
0.7750 
266,505 
0.8150 
266,505  
5 
5 
5 

0 
23 
23 

Unrestricted 
266,505 
232,456 
230,764  
Log Like 