CHAPTER 3. BAYESIAN SPATIAL AUTOREGRESSIVE MODELS110
results.d0
= d0 prior parameter
results.time = time taken for sampling
results.accept= acceptance rate
results.lmax = 1/max eigenvalue of W (or lmax if input)
results.lmin = 1/min eigenvalue of W (or lmin if input)
As the other functions are quite similar, we leave it to the reader to ex-
amine the documentation and demonstration les for these functions. One
point to note regarding use of these functions is that two options exist for
specifying a value for the hyperparameter r. The default is to rely onan im-
proper prior based on r = 4. The other optionallows a proper Gamma(m,k)
prior to be assigned for r.
The rst option has the virtue that convergence will be quicker and less
draws are requiredto produce estimates. The drawbackis that the estimates
are conditional on the single value of r set for the hyperparameter. The
second approach produces draws from a Gamma(m,k) distribution for r on
each pass through the sampler. This produces estimates that average over
alternative r values, in essence integrating over this parameter, resulting in
unconditional estimates.
My experience is that estimates produced with a Gamma(8,2) prior hav-
ing a mean of r = 4 and variance of 2 are quite similar to those based on an
improper prior withr = 4. Use of the Gamma prior tends to require a larger
number of draws based on convergence diagnostics routines implemented by
the function coda from the Econometrics Toolbox described in Chapter 5
of the manual.
3.4 Applied examples
We turn attention to some applications involving the use of these models.
First we present example 3.5 that generates SEM models for a set of 
parameters ranging from 0.1 to 0.9, based on the spatial weight matrix from
the Columbus neighborhood crime data set. Both maximum likelihood and
Gibbs estimates are produced by the program and a table is printed out to
compare the estimation results. The hyperparameter r was set to 30 in this
example, which shouldproduce estimatessimilar to the maximum likelihood
results.
During the loop over alternative data sets, we recover the estimates and
other information we are interestedinfrom the results structures. These are
stored in a matrix that we print using the mprint function to add column
and row labels.
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CHAPTER 3. BAYESIAN SPATIAL AUTOREGRESSIVE MODELS111
% ----- Example 3.5 Using the sem_g function
load wmat.dat;
% standardized 1st-order contiguity matrix
load anselin.dat; % load Anselin (1988) Columbus neighborhood crime data
y = anselin(:,1); n = length(y);
x = [ones(n,1) anselin(:,2:3)];
W = wmat; IN = eye(n);
vnames = strvcat('crime','const','income','house value');
tt = ones(n,1); tt(25:n,1) = [1:25]';
rvec = 0.1:.1:.9; b = ones(3,1);
nr = length(rvec); results = zeros(nr,6);
ndraw = 1100; nomit = 100; prior.rval = 30; bsave = zeros(nr,6);
for i=1:nr, rho = rvec(i);
u = (inv(IN-rho*W))*randn(n,1);
y = x*b + u;
% do maximum likelihood for comparison
resml = sem(y,x,W); prt(resml);
results(i,1) = resml.lam;
results(i,2) = resml.tstat(4,1);
bsave(i,1:3) = resml.beta';
% call Gibbs sampling function
result = sem_g(y,x,W,ndraw,nomit,prior); prt(result);
results(i,3) = mean(result.pdraw);
results(i,4) = results(i,3)/std(result.pdraw);
results(i,5) = result.time;
results(i,6) = result.accept;
bsave(i,4:6) = mean(result.bdraw);
end;
in.rnames = strvcat('True lam','0.1','0.2','0.3', ...
'0.4','0.5','0.6','0.7','0.8','0.9');
in.cnames = strvcat('ML lam','lam t','Gibbs lam','lam t', ...
'time','accept');
mprint(results,in);
in2.cnames = strvcat('b1 ML','b2 ML','b3 ML',...
'b1 Gibbs','b2 Gibbs','b3 Gibbs');
mprint(bsave,in2);
From the results we see that 1100 draws took around 13 seconds. The
acceptance rate falls slightlyfor values of  near 0.9, which we would expect.
Since this is close to the upper limit of unity, we will see an increase in
rejections of candidate values for  that lie outside the feasible range.
The estimates are reasonably similar to the maximum likelihood results
|even for the relatively small number of draws used. In addition to pre-
senting estimates for , we also provide estimates for the parameters  in
the problem.
% sem model demonstration
True lam
ML lam
lam t Gibbs lam
lam t
time
accept
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CHAPTER 3. BAYESIAN SPATIAL AUTOREGRESSIVE MODELS112
0.1
-0.6357
-3.1334
-0.5049
-2.5119
12.9968
0.4475
0.2
0.0598
0.3057
0.0881
0.4296
12.8967
0.4898
0.3
0.3195
1.8975
0.3108
1.9403
13.0212
0.4855
0.4
0.2691
1.5397
0.2091
1.1509
12.9347
0.4827
0.5
0.5399
4.0460
0.5141
3.6204
13.1345
0.4770
0.6
0.7914
10.2477
0.7466
7.4634
13.3044
0.4616
0.7
0.5471
4.1417
0.5303
3.8507
13.2014
0.4827
0.8
0.7457
8.3707
0.7093
6.7513
13.5251
0.4609
0.9
0.8829
17.5062
0.8539
14.6300
13.7529
0.4349
b1 ML
b2 ML
b3 ML
b1 Gibbs
b2 Gibbs
b3 Gibbs
1.0570
1.0085
0.9988
1.0645
1.0112
0.9980
1.2913
1.0100
1.0010
1.2684
1.0121
1.0005
0.7910
1.0298
0.9948
0.7876
1.0310
0.9936
1.5863
0.9343
1.0141
1.5941
0.9283
1.0157
1.2081
0.9966
1.0065
1.2250
0.9980
1.0058
0.3893
1.0005
1.0155
0.3529
1.0005
1.0171
1.2487
0.9584
1.0021
1.3191
0.9544
1.0029
1.8094
1.0319
0.9920
1.8366
1.0348
0.9918
-0.9454
1.0164
0.9925
-0.9783
1.0158
0.9923
As another example, we use a generated data set where we insert 2
outliers. The program generates a vector y based on the Columbus neigh-
borhood crime data set and then adjusts two of the generated y values for
observations 10 and 40 to create outliers.
Example 3.6 produces maximum likelihood and two sets of Bayesian
SAR model estimates. One Bayesian model uses a homoscedastic prior and
the other sets r = 4, creating a a heteroscedastic prior. This is to illustrate
that the dierences in the parameter estimates are due to robustication,
not the Gibbs sampling approach to estimation. A point to consider is that
maximum likelihood estimates of precision based on the informationmatrix
rely on normality, which is violated by the existence of outliers. These
create a disturbance distribution that contains `fatter tails' than the normal
distribution,not unlike the t−distribution. In fact,this is the motivationfor
Geweke's approach to robustifying against outliers. Gibbs estimates based
on a heteroscedastic prior don't rely on normality. If you nd a dierence
between the estimates of precision from maximum likelihood estimates and
Bayesian Gibbs estimates, it is a good indication that outliers may exist.
% ----- Example 3.6 An outlier example
load anselin.dat; load wmat.dat;
load anselin.dat; % load Anselin (1988) Columbus neighborhood crime data
x = [anselin(:,2:3)]; [n k] = size(x); x = [ones(n,1) x];
W = wmat; IN = eye(n);
rho = 0.5;
% true value of rho
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CHAPTER 3. BAYESIAN SPATIAL AUTOREGRESSIVE MODELS113
b = ones(k+1,1); % true value of beta
Winv = inv(IN-rho*W);
y = Winv*x*b + Winv*randn(n,1);
vnames = strvcat('y-simulated','constant','income','house value');
% insert outliers
y(10,1) = y(10,1)*2; y(40,1) = y(40,1)*2;
% do maximum likelihood for comparison
resml = sar(y,x,W); prt(resml,vnames);
ndraw = 1100; nomit = 100;
prior.rval = 100; % homoscedastic model,
resg = sar_g(y,x,W,ndraw,nomit,prior);
prt(resg,vnames);
prior.rval = 4; % heteroscedastic model,
resg2 = sar_g(y,x,W,ndraw,nomit,prior);
prt(resg2,vnames);
% plot the vi-estimates
plot(resg2.vmean);
xlabel('Observations');
ylabel('mean of V_i draws');
The maximum likelihood SAR estimates along with the two sets of
Bayesian model estimates are shown below. We see that the homoscedastic
Gibbs estimates are similar to the maximum likelihood estimates, demon-
strating that the Gibbs sampling estimation procedure is not responsible
for the dierence in estimates we see between maximum likelihood and the
heteroscedastic Bayesian model. For the case of the heteroscedastic prior
we see much better estimates for both  and . Note that the R−squared
statistic is lower for the robust estimates which will be the case because
robustication requires that we not attempt to `t' the outlying observa-
tions. This will generally lead to a worse t for models that produce robust
estimates.
Spatial autoregressive Model Estimates
Dependent Variable =
y-simulated
R-squared
=
0.6779
Rbar-squared
=
0.6639
sigma^2
= 680.8004
Nobs, Nvars
=
49,
3
log-likelihood =
-212.59468
# of iterations =
14
min and max rho =
-1.5362,
1.0000
***************************************************************
Variable
Coefficient
t-statistic
t-probability
constant
19.062669
1.219474
0.228880
income
-0.279572
-0.364364
0.717256
house value
1.966962
8.214406
0.000000
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CHAPTER 3. BAYESIAN SPATIAL AUTOREGRESSIVE MODELS114
rho
0.200210
1.595514
0.117446
Gibbs sampling spatial autoregressive model
Dependent Variable =
y-simulated
R-squared
=
0.6770
sigma^2
= 1077.9188
r-value
=
100
Nobs, Nvars
=
49,
3
ndraws,nomit
=
1100,
100
acceptance rate =
0.9982
time in secs
=
28.9612
min and max rho =
-1.5362,
1.0000
***************************************************************
Variable
Prior Mean
Std Deviation
constant
0.000000
1000000.000000
income
0.000000
1000000.000000
house value
0.000000
1000000.000000
***************************************************************
Posterior Estimates
Variable
Coefficient
t-statistic
t-probability
constant
18.496993
1.079346
0.286060
income
-0.123720
-0.142982
0.886929
house value
1.853332
10.940066
0.000000
rho
0.219656
1.589213
0.118863
Gibbs sampling spatial autoregressive model
Dependent Variable =
y-simulated
R-squared
=
0.6050
sigma^2
= 1374.8421
r-value
=
3
Nobs, Nvars
=
49,
3
ndraws,nomit
=
1100,
100
acceptance rate =
0.9735
time in secs
=
17.2292
min and max rho =
-1.5362,
1.0000
***************************************************************
Variable
Prior Mean
Std Deviation
constant
0.000000
1000000.000000
income
0.000000
1000000.000000
house value
0.000000
1000000.000000
***************************************************************
Posterior Estimates
Variable
Coefficient
t-statistic
t-probability
constant
13.673728
0.778067
0.440513
income
0.988163
0.846761
0.401512
house value
1.133790
3.976679
0.000245
rho
0.388923
2.844965
0.006610
One point that may be of practical importance is that using large values
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CHAPTER 3. BAYESIAN SPATIAL AUTOREGRESSIVE MODELS115
for the hyperparameter r slows down the Gibbs sampling process. This is
because the chi-squared random draws take longer for large r values. This
shows up in this example where the 1100 draws for the homoscedastic prior
based on r = 100 took close to 29 seconds and those for the model based on
r= 4 took only 17.2 seconds. This suggests that a good operationalstrategy
would be not to rely on values of r greater than 30 or 40. These values
may produce v
i
estimates that deviate from unity somewhat, but should in
most cases replicate the maximum likelihood estimates when there are no
outliers.
Abetter strategy is to always rely on a small r value between 2 and 8,
in which case a divergence between the maximum likelihood estimates and
those from the Bayesianmodel reflect the existence of non-constant variance
or outliers.
3.5 An applied exercise
We applied the series of spatial autoregressive models from Section2.5 as
well as corresponding Bayesian spatial autoregressive models to the Boston
data set. Recall that Belsley, Kuh and Welsch (1980) used this data set to
illustrate the impact of outliers and influential observations on least-squares
estimationresults. Here we have an opportunity to see the how the Bayesian
spatial autoregressive models deal with the outliers.
Example 3.7 shows the program code needed to implement both maxi-
mum likelihood and Bayesian models for this data set.
% ----- Example 3.7 Robust Boston model estimation
load boston.raw; % Harrison-Rubinfeld data
load latitude.data; load longitude.data;
[W1 W W3] = xy2cont(latitude,longitude); % create W-matrix
[n k] = size(boston);y = boston(:,k);
% median house values
x = boston(:,1:k-1);
% other variables
vnames = strvcat('hprice','crime','zoning','industry','charlesr', ...
'noxsq','rooms2','houseage','distance','access','taxrate', ...
'pupil/teacher','blackpop','lowclass');
ys = studentize(log(y)); xs = studentize(x);
rmin = 0; rmax = 1;
tic; res1 = sar(ys,xs,W,rmin,rmax); prt(res1,vnames); toc;
prior.rmin = 0; prior.rmax = 1;
prior.rval = 4;
ndraw = 1100; nomit=100;
tic; resg1 = sar_g(ys,xs,W,ndraw,nomit,prior);
prt(resg1,vnames); toc;
tic; res2 = sem(ys,xs,W,rmin,rmax); prt(res2,vnames); toc;
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CHAPTER 3. BAYESIAN SPATIAL AUTOREGRESSIVE MODELS116
tic; resg2 = sem_g(ys,xs,W,ndraw,nomit,prior);
prt(resg2,vnames); toc;
tic; res3 = sac(ys,xs,W,W);
prt(res3,vnames); toc;
tic; resg3 = sac_g(ys,xs,W,W,ndraw,nomit,prior);
prt(resg3,vnames); toc;
An interesting aspect is the timing results which were produced using
the MATLAB `tic' and `toc' commands. Maximum likelihood estimation
of the SAR model took 44 seconds while Gibbs sampling using 1100 draws
and omitting the rst 100 took 124 seconds. For the SEM model the cor-
responding times were 59 and 164 seconds and for the SAC model 114 and
265 seconds. These times seem quite reasonable for this moderately sized
problem.
The results are shown below. (We eliminated the printed output showing
the prior means and standard deviations because all of the Bayesian models
were implemented with diuse priors for the parameters  in the model.) A
prior value of r = 4 was used to produce robustication against outliers and
non-constant variance.
What do we learn from this exercise? First, the parameters  and  for
the SAR and SEM models from maximum likelihood and Bayesian models
are in agreement regarding both the magnitude and signicance. For the
SAC model, the Bayesian estimate for  is in agreement with the maximum
likelihood estimate, but that for  is not. The Bayesian estimate is 0.107
versus 0.188 for maximum likelihood. The maximum likelihood estimate is
signicant whereas the Bayesian estimate is not. This would impact on our
decision regarding which model represents the best specication.
Most of the  estimates are remarkably similar with one notable excep-
tion, that for the `noxsq'pollution variable. Inallthree Bayesianmodels this
estimate is smaller than the maximum likelihood estimate and insignicant
at the 95% level. All maximum likelihood estimates indicate signicance.
This would represent an important policy dierence in the inference made
regarding the impact of air pollution on housing values.
Spatial autoregressive Model Estimates (elapsed_time = 44.2218)
Dependent Variable =
hprice
R-squared
=
0.8421
Rbar-squared
=
0.8383
sigma^2
=
0.1576
Nobs, Nvars
=
506,
13
log-likelihood =
-85.099051
# of iterations =
9
min and max rho =
0.0000,
1.0000
***************************************************************
CHAPTER 3. BAYESIAN SPATIAL AUTOREGRESSIVE MODELS117
Variable
Coefficient
t-statistic
t-probability
crime
-0.165349
-6.888522
0.000000
zoning
0.080662
3.009110
0.002754
industry
0.044302
1.255260
0.209979
charlesr
0.017156
0.918665
0.358720
noxsq
-0.129635
-3.433659
0.000646
rooms2
0.160858
6.547560
0.000000
houseage
0.018530
0.595675
0.551666
distance
-0.215249
-6.103520
0.000000
access
0.272237
5.625288
0.000000
taxrate
-0.221229
-4.165999
0.000037
pupil/teacher
-0.102405
-4.088484
0.000051
blackpop
0.077511
3.772044
0.000182
lowclass
-0.337633
-10.149809
0.000000
rho
0.450871
12.348363
0.000000
Gibbs sampling spatial autoregressive model (elapsed_time = 126.4168)
Dependent Variable =
hprice
R-squared
=
0.8338
sigma^2
=
0.1812
r-value
=
4
Nobs, Nvars
=
506,
13
ndraws,nomit
=
1100,
100
acceptance rate =
0.9910
time in secs
= 110.2646
min and max rho =
0.0000,
1.0000
***************************************************************
Posterior Estimates
Variable
Coefficient
t-statistic
t-probability
crime
-0.127092
-3.308035
0.001008
zoning
0.057234
1.467007
0.143012
industry
0.045240
0.950932
0.342105
charlesr
0.006076
0.249110
0.803379
noxsq
-0.071410
-1.512866
0.130954
rooms2
0.257551
5.794703
0.000000
houseage
-0.031992
-0.748441
0.454551
distance
-0.171671
-5.806800
0.000000
access
0.173901
2.600740
0.009582
taxrate
-0.202977
-5.364128
0.000000
pupil/teacher
-0.086710
-3.081886
0.002172
blackpop
0.094987
3.658802
0.000281
lowclass
-0.257394
-5.944543
0.000000
rho
0.479126
5.697191
0.000000
Spatial error Model Estimates
(elapsed_time = 59.0996)
Dependent Variable =
hprice
R-squared
=
0.8708
Rbar-squared
=
0.8676
sigma^2
=
0.1290
CHAPTER 3. BAYESIAN SPATIAL AUTOREGRESSIVE MODELS118
log-likelihood =
-58.604971
Nobs, Nvars
=
506,
13
# iterations
=
10
min and max lam =
0.0000,
1.0000
***************************************************************
Variable
Coefficient
t-statistic
t-probability
crime
-0.186710
-8.439402
0.000000
zoning
0.056418
1.820113
0.069348
industry
-0.000172
-0.003579
0.997146
charlesr
-0.014515
-0.678562
0.497734
noxsq
-0.220228
-3.683553
0.000255
rooms2
0.198585
8.325187
0.000000
houseage
-0.065056
-1.744224
0.081743
distance
-0.224595
-3.421361
0.000675
access
0.352244
5.448380
0.000000
taxrate
-0.257567
-4.527055
0.000008
pupil/teacher
-0.122363
-3.839952
0.000139
blackpop
0.129036
4.802657
0.000002
lowclass
-0.380295
-10.625978
0.000000
lambda
0.757669
19.133467
0.000000
Gibbs sampling spatial error model (elapsed_time = 164.8779)
Dependent Variable =
hprice
R-squared
=
0.7313
sigma^2
=
0.1442
r-value
=
4
Nobs, Nvars
=
506,
13
ndraws,nomit
=
1100,
100
acceptance rate
=
0.4715
time in secs
= 116.2418
min and max lambda =
-1.9826,
1.0000
***************************************************************
Posterior Estimates
Variable
Coefficient
t-statistic
t-probability
crime
-0.165360
-3.967705
0.000083
zoning
0.048894
1.226830
0.220472
industry
-0.002985
-0.051465
0.958976
charlesr
-0.014862
-0.538184
0.590693
noxsq
-0.145616
-1.879196
0.060807
rooms2
0.339991
7.962844
0.000000
houseage
-0.130692
-2.765320
0.005900
distance
-0.175513
-2.398220
0.016846
access
0.276588
3.121642
0.001904
taxrate
-0.234511
-4.791976
0.000002
pupil/teacher
-0.085891
-2.899236
0.003908
blackpop
0.144119
4.773623
0.000002
lowclass
-0.241751
-5.931583
0.000000
lambda
0.788149
19.750640
0.000000
CHAPTER 3. BAYESIAN SPATIAL AUTOREGRESSIVE MODELS119
General Spatial Model Estimates (elapsed_time = 114.5267)
Dependent Variable =
hprice
R-squared
=
0.8662
Rbar-squared
=
0.8630
sigma^2
=
0.1335
log-likelihood =
-55.200525
Nobs, Nvars
=
506,
13
# iterations
=
7
***************************************************************
Variable
Coefficient
t-statistic
t-probability
crime
-0.198184
-8.766862
0.000000
zoning
0.086579
2.824768
0.004923
industry
0.026961
0.585884
0.558222
charlesr
-0.004154
-0.194727
0.845687
noxsq
-0.184557
-3.322769
0.000958
rooms2
0.208631
8.573808
0.000000
houseage
-0.049980
-1.337513
0.181672
distance
-0.283474
-5.147088
0.000000
access
0.335479
5.502331
0.000000
taxrate
-0.257478
-4.533481
0.000007
pupil/teacher
-0.120775
-3.974717
0.000081
blackpop
0.126116
4.768082
0.000002
lowclass
-0.374514
-10.707764
0.000000
rho
0.625963
9.519920
0.000000
lambda
0.188257
3.059010
0.002342
Gibbs sampling general spatial model (elapsed_time = 270.8657)
Dependent Variable =
hprice
R-squared
=
0.7836
sigma^2
=
0.1487
r-value
=
4
Nobs, Nvars
=
506,
13
ndraws,nomit
=
1100,
100
accept rho rate
=
0.8054
accept lam rate
=
0.9985
time in secs
= 205.6773
min and max rho
=
0.0000,
1.0000
min and max lambda =
-1.9826,
1.0000
***************************************************************
Posterior Estimates
Variable
Coefficient
t-statistic
t-probability
crime
-0.161673
-3.905439
0.000107
zoning
0.047727
1.274708
0.203013
industry
0.024687
0.433283
0.664999
charlesr
0.008255
0.319987
0.749114
noxsq
-0.121782
-1.814226
0.070251
rooms2
0.333823
7.332684
0.000000
houseage
-0.098357
-2.080260
0.038018
distance
-0.193060
-3.798247
0.000164
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