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CHAPTER 2. SPATIAL AUTOREGRESSIVE MODELS

500

1000

1500

2000

2500

3000

500

1000

1500

2000

2500

3000

nz = 12428

Figure 2.2: Sparsity structure of W from Pace and Barry

even ner grid with increments of 0.001 around the optimal estimate from

the second pass.

Note that we used a MATLAB sparse function eigs to solve for the

eigenvalues of the contiguity matrix W, which requires 60 seconds to solve

this part of the problem as shown in the output below. The time necessary

to perform each pass over the grid of 21 values for  was around 10 seconds.

With a total of 3 passes to produce an estimate of  accurate to 3 decimal

digits, we have a total elapsed time of 1 minute and 30 seconds to solve

for the maximum likelihood estimate of . This is certainly a reasonable

amount of computational time for such a large problem on a reasonably in-

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CHAPTER 2. SPATIAL AUTOREGRESSIVE MODELS

expensive desktop computing platform. Of course, there is still the problem

of producing measures of dispersion for the estimates that Pace and Barry

address by suggesting the use of likelihood ratio statistics.

elapsed_time = 59.8226 % computing min,max eigenvalues

elapsed_time = 10.5280 % carrying out 1st 21-point grid over rho

elapsed_time = 10.3791 % carrying out 2nd 21-point grid over rho

elapsed_time = 10.3747 % carrying out 3rd 21-point grid over rho

estimate of rho =

0.7220

estimate of sigma =

0.0054

How does our approach compare to that of Pace and Barry? Example2.3

shows a program to estimate the same FAR model using our far function.

% ----- Example 2.3 Using the far() function

with very large data set from Pace and Barry

load elect.dat;

% load data on votes

y = elect(:,7)./elect(:,8); % proportion of voters casting votes

ydev = y - mean(y);

% deviations from the means form

clear y;

% conserve on RAM memory

clear elect; % conserve on RAM memory

load ford.dat; % 1st order contiguity matrix stored in sparse matrix form

ii = ford(:,1); jj = ford(:,2); ss = ford(:,3);

n = 3107;

clear ford; % clear ford matrix to save RAM memory

W = sparse(ii,jj,ss,n,n);

clear ii; clear jj; clear ss; % conserve on RAM memory

tic; res = far(ydev,W); toc;

prt(res);

In terms of time needed to solve the problem, our use of the simplex

optimization algorithm takes only 10.6 seconds to produce a more accurate

estimate than that based on the grid approach of Pace and Barry. Their

approach which we modied took 30 seconds to solve for a  value accurate

to 3 decimal digits. Note also in contrast to Pace and Barry, we compute

aconventional measure of dispersion using the numerical hessian estimates

which takes only 1.76 seconds. The total time required to compute not only

the estimates and measures of dispersion for  and , but the R−squared

statistics and log likelihood function was around 100 seconds.

elapsed_time = 59.8226 % computing min,max eigenvalues

elapsed_time = 10.6622 % time required for simplex solution of rho

elapsed_time = 1.7681 % time required for hessian evaluation

elapsed_time = 1.7743 % time required for likelihood evaluation

total time

= 74.01

% comparable time to Pace and Barry

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CHAPTER 2. SPATIAL AUTOREGRESSIVE MODELS

First-order spatial autoregressive model Estimates

R-squared

0.5375

sigma^2

0.0054

Nobs, Nvars

3107,

log-likelihood =

1727.9824

# of iterations =

min and max rho =

-1.0710,

1.0000

***************************************************************

Variable

Coefficient

t-statistic

t-probability

rho

0.721474

59.495159

0.000000

Many of the ideas developed in this section regarding the use of MAT-

LAB sparse matrix algorithms will apply equally to the estimation proce-

dures we develop in the next three sections for the other members of the

spatial autoregressive model family.

2.2 The mixed autoregressive-regressive model

This model extends the rst-order spatial autoregressive model to include

amatrix X of explanatory variables such as those used in traditional re-

gression models. Anselin (1988) provides a maximum likelihood method for

estimating the parameters of this model that he labels a `mixed regressive

-spatial autoregressive model'. We will refer to this model as the spatial

autoregressive model (SAR). The SAR model takes the form:

y = Wy + X +"

(2.12)

"  N(0;

)

Where y contains an nx1 vector of dependent variables, X represents the

usual nxk data matrix containing explanatory variables and W is a known

spatialweight matrix,usually a rst-order contiguitymatrix. The parameter

is a coecient on the spatially lagged dependent variable, Wy, and the

parameters  reﬂect the inﬂuence of the explanatory variables on variation

in the dependent variable y. The model is termed a mixed regressive -

spatial autoregressive model because it combines the standard regression

model with a spatially lagged dependent variable, reminiscent of the lagged

dependent variable model from time-series analysis.

Maximum likelihood estimationof this model is based on a concentrated

likelihood function as was the case with the FAR model. A few regressions

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CHAPTER 2. SPATIAL AUTOREGRESSIVE MODELS

are carried out along with a univariate parameter optimization of the con-

centrated likelihood function over values of the autoregressive parameter .

The steps are enumerated in Anselin (1988) as:

1. perform OLS for the model: y = X

2. perform OLS for the model Wy = X

3. compute residuals e

=y− X



and e

=Wy −X



4. given e

and e

nd  that maximizes the concentrated likelihood

function: L

=C − (n=2)ln(1=n)(e

−e

)

−e

)+lnjI − Wj

5. given ^ that maximizes L

, compute

 = (



−



) and ^

(1=n)(e

−e

)

−e

)

Again we face the problem that using a univariate simplex optimization

algorithm to nd a maximum likelihood estimate of  based on the concen-

trated log likelihood function leaves us with no estimates of the dispersion

associated with the parameters. We can overcome this using the theoretical

information matrix for small problems and the numerical hessian approach

introduced for the FAR model in the case of large problems.

Since this model is quite similar to the FAR model which we already

presented, we will turn immediately to describing the function.

2.2.1 The sar() function

The function sar is fairlysimilar to our far function,withthedocumentation

presented below.

PURPOSE: computes spatial autoregressive model estimates

y = p*W*y + X*b + e, using sparse matrix algorithms

---------------------------------------------------

USAGE: results = sar(y,x,W,rmin,rmax,convg,maxit)

where: y = dependent variable vector

x = explanatory variables matrix

W = standardized contiguity matrix

rmin = (optional) minimum value of rho to use in search

rmax = (optional) maximum value of rho to use in search

convg = (optional) convergence criterion (default = 1e-8)

maxit = (optional) maximum # of iterations (default = 500)

---------------------------------------------------

RETURNS: a structure

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CHAPTER 2. SPATIAL AUTOREGRESSIVE MODELS

results.meth = 'sar'

results.beta = bhat

results.rho

= rho

results.tstat = asymp t-stat (last entry is rho)

results.yhat = yhat

results.resid = residuals

results.sige = sige = (y-p*W*y-x*b)'*(y-p*W*y-x*b)/n

results.rsqr = rsquared

results.rbar = rbar-squared

results.lik

= -log likelihood

results.nobs = # of observations

results.nvar = # of explanatory variables in x

results.y

= y data vector

results.iter

= # of iterations taken

results.romax = 1/max eigenvalue of W (or rmax if input)

results.romin = 1/min eigenvalue of W (or rmin if input)

--------------------------------------------------

As in the case of far, we allow the user to provide minimum and maxi-

mum values of  to use in the search. This may save time in cases where we

wish to restrict our estimate of  to the positive range.

The other point to note is that this function also uses the numerical

hessian approach to compute measures of dispersion for large problems in-

volving more than 500 observations.

2.2.2 Applied examples

As anillustrationof using the sar function, consider the programin example

2.4, where we estimate a model to explain variation in votes casts on a

per capita basis in the 3,107 counties. The explanatory variables in the

model were: the proportion of population with high school level education

or higher, the proportion of the population that are homeowners and the

income per capita. Note that the population deﬂater used to convert the

variables to per capita terms was the population 18 years or older in the

county.

% ----- Example 2.4 Using the sar() function with a very large data set

load elect.dat;

% load data on votes in 3,107 counties

y = (elect(:,7)./elect(:,8));

% convert to per capita variables

x1 = log(elect(:,9)./elect(:,8)); % education

x2 = log(elect(:,10)./elect(:,8));% homeownership

x3 = log(elect(:,11)./elect(:,8));% income

n = length(y); x = [ones(n,1) x1 x2 x3];

clear x1; clear x2; clear x3;

clear elect;

% conserve on RAM memory

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CHAPTER 2. SPATIAL AUTOREGRESSIVE MODELS

load ford.dat; % 1st order contiguity matrix stored in sparse matrix form

ii = ford(:,1); jj = ford(:,2); ss = ford(:,3);

n = 3107;

clear ford; % clear ford matrix to save RAM memory

W = sparse(ii,jj,ss,n,n);

clear ii; clear jj; clear ss; % conserve on RAM memory

vnames = strvcat('voters','const','educ','homeowners','income');

to = clock;

res = sar(y,x,W);

etime(clock,to)

prt(res,vnames);

We use the MATLAB clock function as well as etime to determine the

overall execution time needed to solve this problem, which was 130 seconds.

The estimation results are presented below:

Spatial autoregressive Model Estimates

Dependent Variable =

voters

R-squared

0.6419

Rbar-squared

0.6416

sigma^2

0.0042

Nobs, Nvars

3107,

log-likelihood =

5053.5179

# of iterations =

min and max rho =

-1.0710,

1.0000

***************************************************************

Variable

Coefficient

t-statistic

t-probability

const

0.753169

25.963031

0.000000

educ

0.148553

17.341543

0.000000

homeowners

0.208960

26.142340

0.000000

income

-0.085462

-9.413244

0.000000

rho

0.563764

39.797104

0.000000

We see from the results that all of the explanatory variables exhibit a

signicant eect on the variable we wished to explain. The results also

indicate that the dependent variable y exhibits strong spatial dependence

even after taking the eect of these variables into account as the estimate

of  on the spatial lagged variable is large and signicant.

As an illustration of the bias associated with least-squares estimation

of spatial autoregressive models, we present an example based on a spatial

sample of 88 observations for counties in the state of Ohio. A sample of

average housing values for each of 88 counties in Ohio will be related to

population per square mile, the housing density and unemployment rates in

each county. This regression relationship can be written as:

HOUSE

=+ POP

+γHDENSITY

+UNEMPLOY

(2.13)

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CHAPTER 2. SPATIAL AUTOREGRESSIVE MODELS

The motivation for the regression relationship is that population and

household density as well as unemployment rates work to determine the

house values in each county. Consider that the advent of suburban sprawl

and the notion of urban rent gradients suggests that housing values in con-

tiguous counties should be related. The least-squares relationship in (2.13)

ignores the spatial contiguity information whereas the SAR model would

allow for this type of variation in the model.

The rst task is to construct a spatial contiguity matrix for use with

our spatial autoregressive model. This could be accomplished by examin-

ing a map of the 88 counties and recording neighboring tracts for every

observation, a very tedious task. An alternative is to use the latitude and

longitude coordinates to construct a contiguity matrix. We rely on a func-

tion xy2cont that carries out this task. This function is part of Pace and

Barry's Spatial Statistics Toolbox for MATLAB, but has been modied to

t the documentation conventions of the spatial econometrics library. The

function documentation is shown below:

PURPOSE: uses x,y coord to produce spatial contiguity weight matrices

with delaunay routine from MATLAB version 5.2

------------------------------------------------------

USAGE: [w1 w2 w3] = xy2cont(xcoord,ycoord)

where:

xcoord = x-direction coordinate vector (nobs x 1)

ycoord = y-direction coordinate vector (nobs x 1)

------------------------------------------------------

RETURNS: w1 = W*W*S, a row-stochastic spatial weight matrix

w2 = W*S*W, a symmetric spatial weight matrix (max(eig)=1)

w3 = diagonal matrix with i,i equal to 1/sqrt(sum of ith row)

------------------------------------------------------

References: Kelley Pace, Spatial Statistics Toolbox 1.0

------------------------------------------------------

This function essentially uses triangles connecting the x-y coordinates in

space to deduce contiguous entities. As an example of using the function,

consider constructing a spatial contiguitymatrix for the Columbus neighbor-

hood crime data set where we know both the rst-order contiguity structure

taken from a map of the neighborhoods as well as the x-y coordinates. Here

is a program to generate the rst-order contiguity matrix from the latitude

and longitude coordinates and produce a graphical comparison of the two

contiguity structures shown in Figure2.3.

% ----- Example 2.5 Using the xy2cont() function

load anselin.data; % Columbus neighborhood crime

xc = anselin(:,5); % longitude coordinate

yc = anselin(:,4); % latitude coordinate

CHAPTER 2. SPATIAL AUTOREGRESSIVE MODELS

load Wmat.data;

% load standardized contiguity matrix

% create contiguity matrix from x-y coordinates

[W1 W2 W3] = xy2cont(xc,yc);

% graphically compare the two

spy(W2,'ok'); hold on; spy(Wmat,'+k');

legend('generated','actual');

nz = 232

generated

actual

Figure 2.3: Generated contiguity structure results

Example2.6reads in the data fromtwoles containinga database for the

88 Ohio counties as wellas data vectors containingthe latitudeand longitude

informationneeded to construct a contiguity matrix. We rely on a log trans-

formationof thedependent variablehouse values toprovide better scaling for

the data. Note the use of the MATLAB construct: `ohio2(:,5)./ohio1(:,2)',

CHAPTER 2. SPATIAL AUTOREGRESSIVE MODELS

which divides every element in the column vector `ohio(:,5)'containing total

households in eachcounty by every element inthe column vector `ohio1(:,2)',

which containsthe populationfor every county. This produces the number of

households per capita for each county as an explanatory variable measuring

household density.

% ----- Example 2.6 Least-squares bias

demonstrated with Ohio county data base

load ohio1.dat; % 88 counties (observations)

% 10 columns

% col1 area in square miles

% col2 total population

% col3 population per square mile

% col4 black population

% col5 blacks as a percentage of population

% col6 number of hospitals

% col7 total crimes

% col8 crime rate per capita

% col9 population that are high school graduates

% col10 population that are college graduates

load ohio2.dat; % 88 counties

% 10 columns

% col1 income per capita

% col2 average family income

% col3 families in poverty

% col4 percent of families in poverty

% col5 total number of households

% col6 average housing value

% col7 unemployment rate

% col8 total manufacturing employment

% col9 manufacturing employment as a percent of total

% col10 total employment

load ohio.xy; % latitude-longitude coordinates of county centroids

[junk W junk2] = xy2cont(ohio(:,1),ohio(:,2)); % make W-matrix

y = log(ohio2(:,6)); n = length(y);

x = [ ones(n,1) ohio1(:,3) ohio2(:,5)./ohio1(:,2) ohio2(:,7) ];

vnames = strvcat('hvalue','constant','popsqm','housedensity','urate');

res = ols(y,x);

prt(res,vnames);

res = sar(y,x,W); prt(res,vnames);

The results from these two regressions are shown below. The rst point

to note is that the spatial autocorrelation coecient estimate for the SAR

model is statistically signicant, indicating the presence of spatial autocor-

relation in the regression relationship. Least-squares ignores this type of

variation producing estimates that lead us to conclude that all three ex-

planatory variables are signicant in explaining housing values across the 88

county sample. In contrast, the SAR model leads us to conclude that the

CHAPTER 2. SPATIAL AUTOREGRESSIVE MODELS

population density (popsqm) is not statistically signicant at conventional

levels. Keep in mind that the OLS estimates are biased and inconsistent,

so the inference of signicance from OLS we would draw is likely to be

incorrect.

Ordinary Least-squares Estimates

Dependent Variable =

hvalue

R-squared

0.6292

Rbar-squared

0.6160

sigma^2

0.0219

Durbin-Watson =

2.0992

Nobs, Nvars

88,

***************************************************************

Variable

Coefficient

t-statistic

t-probability

constant

11.996858

71.173358

0.000000

popsqm

0.000110

2.983046

0.003735

housedensity

-1.597930

-3.344910

0.001232

urate

-0.067693

-7.525022

0.000000

Spatial autoregressive Model Estimates

Dependent Variable =

hvalue

R-squared

0.7298

Rbar-squared

0.7201

sigma^2

0.0153

Nobs, Nvars

88,

log-likelihood =

87.284225

# of iterations =

min and max rho =

-2.0158,

1.0000

***************************************************************

Variable

Coefficient

t-statistic

t-probability

constant

6.300144

35.621170

0.000000

popsqm

0.000037

1.196689

0.234794

housedensity

-1.251435

-3.140028

0.002332

urate

-0.055474

-7.387845

0.000000

rho

0.504131

53.749348

0.000000

Asecond point is that taking the spatial variation into account improves

the t of the model, raising the R-squared statistic for the SAR model.

Finally, the magnitudes of the OLS parameter estimates indicate that house

values are more sensitive to the household density and the unemployment

rate variables than the SAR model. For example, the OLS estimates imply

that a one percentage point increase in the unemployment rate leads to

adecrease of 6.8 percent in house values whereas the SAR model places

this at 5.5 percent. Similarly, the OLS estimates for household density is

considerably larger in magnitude than that from the SAR model.

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