The implications of ignoring potential spatial dependence in county-level yield data are discussed. Spatial dependence in a county-level yield data set is identified and methods for correcting the dependence via spatial weighting matrices and generalized least squares regression are performed. The paper also examines how the spatial dependence declines as the distance between observations increases.Productivity Analysis, Research Methods/ Statistical Methods,

DiRienzo, Cassandra

Fackler, Paul L.

Goodwin, Barry K.

English

Research Papers in Economics

Modeling Spatial Dependence and Spatial Heterogeneity in 
County Yield Forecasting Models 
 
 
 
 
August 1, 2000 
AAEA Conference  
Tampa, Summer 2000 
Session 8  
 
 
 
 
Cassandra DiRienzo 
Paul Fackler 
Barry K. Goodwin 
North Carolina State University 
 
 
 
Abstract 
The implications of ignoring potential spatial dependence in county-level yield data are 
discussed.  Spatial dependence in a county-level yield data set is identified and methods 
for correcting the dependence via spatial weighting matrices and generalized least 
squares regression are performed.  The paper also examines how the spatial dependence 
declines as the distance between observations increases.  
 
 
 
 
Modeling Spatial Dependence and Spatial Heterogeneity in County Yield 
Forecasting Models 
 
1.  Introduction 
  Econometric analyses using cross-sectional or panel data often face unique 
challenges.  In particular, there are two issues that often arise in sample data sets 
containing a locational or spatial element: spatial dependence and spatial heterogeneity.  
Spatial dependence occurs when an observation recorded at one point in space is 
dependent on an observation(s) at another point(s) in space.  Spatial heterogeneity occurs 
when there is variation in the relationship between the observations through space.  Both 
issues can occur whenever a data set is collected according to a locational or spatial 
element such as county, state, zip code, or other similar methods.  If the data are not 
adjusted to reflect this spatial relationship, the classical assumptions used in regression 
modeling are violated. 
  Recognition of spatial dependence is a problem of particular importance in 
modeling yield trends and yield distributions.  In particular, crop yields at spatially 
distinct locations are likely to be highly correlated if we consider that weather events are 
highly correlated over space.  Ignorance of spatial correlation may result in inefficient 
forecasts or misleading inferences regarding yield distributions.  This issue takes on even 
greater significance if we consider the variety of new crop and revenue insurance 
programs, which require accurate estimation of yield and price distributions, as well as the correlation between prices and yields.  In particular, actuarially fair insurance 
premiums depend on accurate estimates of crop yield distributions.  If crop yields are 
spatially correlated and the correlation is ignored, the estimated crop yield distributions 
will be inaccurate and misleading, and the resulting crop insurance premiums will not be 
actuarially fair. 
Past approaches to estimating yield distributions have over-looked the potential 
existence of spatial dependence (Botts and Boles) by assuming a standard normal 
distribution of crop yields.  More recent applications (see, for example, Goodwin and 
Ker) have used ad-hoc methods to account for the spatial dependence in yield processes.  
To date, there has not been a comprehensive study that estimates and corrects spatial 
dependence in county-level crop yield data. 
Although this analysis is not comprehensive, our objective is to offer methods for 
correcting spatial dependence in county-level yield data and to evaluate county-level 
yield trend models in light of this spatial dependence.  There are several ways to account 
for spatial dependence within a data set.  In this analysis, we define a spatial weighting 
matrix that weights the observations based on their spatial proximity.  We then use the 
spatial weighting matrix to transform the data to reflect the spatial dependence.  Given 
the correct form of the weighting matrix, the data will no longer violate the classical 
assumptions. 
 
2.  The Data Set 
  The panel data set we have used in this analysis is the county-level corn yields per 
harvested acre in Iowa ranging from 1958 to 1997.  There are a total of ninety-nine counties in Iowa that report annual corn yields per harvested acre.  This produces a panel 
data set consisting of ninety-nine observations per year over forty years.  Given that the 
county-level yields were collected by their respective FIPS code, there is potentially a 
significant spatial dependence problem in this data set.  It is highly likely that the 
reported county-level yields within Iowa are correlated across space if we consider 
similar weather patterns, soil, pest damages, etc.  If we fail to recognize this spatial 
dependence, the typical regression analysis will yield inefficient estimates. 
  To account for the spatial dependence, we defined a spatial weighting matrix. To 
create the spatial weighting matrix, we converted the county FIPS codes into their 
respective centroid latitude and longitude coordinates to determine the Great Circle 
distance (in miles) between each of the counties.  The spatial weighting matrix was 
created by taking the exponential of the negative of the Great Circle distance between 
each county.  We defined this matrix as 
D = exp(-A) 
where A represents a (99x99) symmetric matrix containing the Great Circle Distance 
between each of the ninety-nine counties.  The exponential operator represents the 
exponential of each element in A and not the exponential of the matrix A.  D, therefore, is 
a (99x99) symmetric matrix with ones along the diagonal. 
 
3.  Estimation of the Spatial Dependence Inherent in the Data Set 
Initially, we sought an estimate of the spatial dependence inherent in our data set.  
We followed Anselin’s (1988) approach and formed the following regression 
yt = ñCyt + å t  where yt represents a (99x40) x 1 vector of de-trended corn yields per harvested acre, C is 
a (3960x3960) matrix equal to the Kronecker product of D and a (40x40) Identity matrix, 
and å t is a (99x40) x 1 vector representing the stochastic disturbance. ñ provides an 
estimate of the spatial dependence contained within our data set and will measure the 
average influence of neighboring observations on observations contained in y (LeSage 
1998).  It is important to note that ñ should not be interpreted as a correlation coefficient. 
  We obtained the following estimation results for ñ 
Variable  Estimate  Standard Error  t Value 
ñ  0.98809  0.10504  9.41 
 
  Additionally, ñCy provides an estimate of the variation in y, or the influence on y, 
that can be explained by spatial dependence (LeSage 1998).  The results of this analysis 
can be found in Table 1.  The results indicate that the variation or influence on the de-
trended yields that can be attributed to spatial dependence differ significantly across 
counties.  However, although the amount of variation or influence on y differs across 
counties, each county’s yield is affected by its neighbor’s yield and ignoring this 
dependency will produce misleading results. 
 
4.  Modified Regression Analysis 
  In order not to violate the classical regression assumptions, we need to account for 
the spatial dependence within the data set.  In this case, our model has the following  
form: 
yt = Xâ + å t where yt is the same as described above, X is a set of regressors, and å t is a vector of 
Gaussian residuals with Cov(å) = ó
2D, where D again represents the spatial weighting 
matrix described above.  Therefore, the log-likelihood function for this model is: 
L = nlnó
2 + nln|D| + (1/ ó
2)(y – Xâ)`C
-1(y – Xâ) 
where C is a (3960x3960) matrix equal to the Kronecker product of D and a (40x40) 
Identity matrix.  The maximum likelihood estimators for â and ó
2 are: 
âmle = (X`C
-1X)
-1X` C
-1yt 
ó
2
mle = (å t` C
-1 å t)/n 
The above estimators take into consideration the spatial dependence contained within the 
data set and will yield unbiased and efficient estimators of the true parameters. 
 
5.  Rate of Spatial Dependence Decay   
  Given that a multitude of data is collected via a locational or spatial component, it 
is important to determine the rate at which the spatial dependence decreases as distance 
between observations increases.  To obtain such an estimate, we defined the following 
model: 
yt = Xâ + å t 
where yt and å t are the same as described above, and X represents a matrix composed of 
two smaller matrices, Z and Z*.  Z represents a (99x40) x 25 matrix of basis coordinates 
and Z* represents a (99x40) x 25 matrix of basis coordinates as a function of time.  The 
basis coordinates represent the latitude and longitude coordinates for each of ninety-nine 
counties fit to a fourth-order polynomial.  Note that this regression yields a vector of 
smoothed estimates, â, which are a function of latitude and longitude.     We now define  
W = exp(-á*A) 
where A (as described above) is a (99x99) symmetric matrix representing the Great 
Circle Distance between each of the ninety-nine counties.  The exponential operator 
represents the exponential of each element contained in A and not the exponential of the 
matrix A and á represents the rate in which the spatial dependence decreases as distance 
increases.  Note that in the previous analysis, we assumed the value of á to be one.  If we 
consider the log-likelihood function above
1 and the maximum likelihood estimator
2 for 
ó
2, the negative of the concentrated log-likelihood function for this model is: 
Lc  = n + ln|W| + nln((å t` M
-1 å t)/nT) 
where M is a (3960x3960) matrix equal to the Kronecker product of W and a (40x40) 
Identity matrix, and, in this case, n is equal to ninety-nine and T is equal to forty.  
  We obtained the following estimate of alpha. 
Alpha  Maximized Value of Lc 
2.8591  99.3810 
 
The above estimate of alpha indicates that spatial dependence decreases at an exponential 
rate as distance between observations increases.  Therefore, we can interpret that the 
observations that are closest in distance to each other have the greatest degree of spatial 
                                                 
1 L = nlnó
2 + nln|D| + (1/ ó
2)(y – Xâ)`C
-1(y – Xâ) where, in this case, alpha was set to one. 
 
2 ó
2
mle = (å t` C
-1 å t)/n where, in this case, alpha was set to one. 
 dependence and this spatial dependence decreases exponentially as the distance between 
the observations increases.   
 
6.  Conclusions 
  As the above analysis indicates, data collected by a locational or spatial element 
can contain spatial dependence, and, if proper adjustments are not made, misleading and 
inefficient estimates will be obtained.  In particular, ignorance of spatial dependence in 
county-level crop yields can have a detrimental impact on crop insurance premiums.  
Ignoring spatial dependence will inflate the variance of the yields, distort the yield 
distribution and produce insurance premiums that are not actuarially fair. Table 1.  Estimate of the Total Variation in Yields Explained by Spatial Dependence 
County FIPS Code  ñDy Estimate 
19001  -0.28341 
19003  -0.00118 
19005  -0.0533 
19007  -0.40134 
19009  -0.07387 
19011  -0.09592 
19013  0.073073 
19015  0.005301 
19017  0.03228 
19019  0.053383 
19021  -0.16908 
19023  0.138884 
19025  -0.1098 
19027  0.024962 
19029  -0.00162 
19031  0.012505 
19033  -0.21775 
19035  0.015699 
19037  -0.29306 
19039  0.044139 
19041  0.017667 County FIPS Code  ñDy Estimate 
19043  -0.02514 
19045  -0.23347 
19047  0.490033 
19049  0.124972 
19051  0.044102 
19053  0.015583 
19055  0.095545 
19057  0.17747 
19059  -0.00033 
19061  0.025785 
19063  -0.05822 
19065  0.081519 
19067  0.178518 
19069  0.227268 
19071  0.542731 
19073  0.222898 
19075  0.617847 
19077  0.312662 
19079  -0.06454 
19081  -0.05263 
19083  -0.58968 
19085  0.066442 County FIPS Code  ñDy Estimate 
19087  -0.15102 
19089  -0.25844 
19091  -0.09045 
19093  0.018276 
19095  -0.10789 
19097  -0.62759 
19099  0.112119 
19101  0.014402 
19103  0.072057 
19105  0.283049 
19107  0.282777 
19109  0.10308 
19111  -0.08162 
19113  0.016237 
19115  0.373224 
19117  0.074019 
19119  0.686339 
19121  -0.04382 
19123  -0.20372 
19125  -0.2773 
19127  -0.01462 
19129  -.165 County FIPS Code  ñDy Estimate 
19131  -0.12418 
19133  0.119638 
19135  0.460725 
19137  0.06961 
19139  0.281481 
19141  0.104043 
19143  0.462193 
19145  0.211104 
19147  -0.04501 
19149  -0.68542 
19151  0.103419 
19153  -0.63501 
19155  -0.09109 
19157  -0.4131 
19159  0.273739 
19161  -0.0406 
19163  -0.00075 
19165  -0.205 
19167  -0.0021 
19169  0.277505 
19171  0.292873 
19173  -0.81781 County FIPS Code  ñDy Estimate 
19175  -0.33346 
19177  -0.31656 
19179  0.529671 
19181  0.246346 
19183  0.009247 
19185  -0.12278 
19187  -0.00358 
19189  0.053965 
19191  0.03662 
19193  -0.13134 
19195  0.023556 
 References 
Anselin, L. 1998. Spatial Econometrics: Methods and Models, (Dorddrecht: Kluwer 
Academic Publishers). 
Amemiya, T.  1985. Advanced Econometrics, (Cambridge, MA: Harvard University 
Press). 
Brudson, C.A. S. Fotheringham, and M. Charlton.  1996.  “Geographically weighted 
regression: a method for exploring spatial nonstationarity,” Geographical Analysis, Vol. 
28, pp. 281-298. 
Casetti, E.,  1972. “Generating Models by the Expansion Method: Applications to 
Geographic Research”, Geographical Analysis, Vol. 4, pp. 81-91. 
Casetti, E.,  1992. “Bayesian Regression and the Expansion Method”, Geographical 
Analysis, Vol. 24, pp. 58-74. 
Cliff, A.D. and J.K. Ord,  1973.  “Spatial Autocorrelation”, Pion Limitied, Brondesbury 
Park, London. 
Goodwin, B and A. Ker, 1998.  “Nonparametric Estimation of Crop Yield Distributions: 
Implication For Rating Group-Risk Crop Insurance Contracts”, American Journal of 
Agricultural Economics, Vol. 80, pp. 139-153. 
LeSage, J.P. 1998.  “Spatial Econometrics”, Unpublished manuscript, Department of 
Economics, University of Toledo. 
McMillen, D.P. 1996.  “One hundred fifty years of land values in Chicago: a 
nonparametric approach,” Journal of Urban Economics, Vol. 40, pp. 100-124. 
Odland, J.  1988.  “Spatial Autocorrelation”, Sage Publications, Scientific Geography 
Series, Vol. 9.  
 
 
 

Advanced Econometrics,

Bayesian Regression and the Expansion Method”,

Generating Models by the Expansion Method: Applications to Geographic Research”,

Geographically weighted regression: a method for exploring spatial nonstationarity,”

Nonparametric Estimation of Crop Yield Distributions: Implication For Rating Group-Risk Crop Insurance Contracts”,

One hundred fifty years of land values in Chicago: a nonparametric approach,”

Spatial Autocorrelation”, Pion Limitied,

Spatial Econometrics: Methods and Models, (Dorddrecht:

MODELING SPATIAL DEPENDENCE AND SPATIAL HETEROGENEITY IN COUNTY YIELD FORECASTING MODELS

MODELING SPATIAL DEPENDENCE AND SPATIAL HETEROGENEITY IN COUNTY YIELD FORECASTING MODELS

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