Journal of Agricultural Economics thn 2009
Journal of Agricultural Economics, Vol. 60, No. 1, 2009, 171–189
doi: 10.1111/j.1477-9552.2008.00182.x
A Spatio-temporal Model of Farmland
Values
David Maddison1
(Original submitted November 2007, revision received August 2008, accepted
September 2008.)
Abstract
Despite the fact that data on farm sales are invariably collected over both time
and space, previous papers have allowed for the presence of either temporal or spatial relationships in the data, but not both. Some papers have also inadvertently
assumed that although farmland values are influenced by prices realised by nearby
sales, these sales need not necessarily be comparable in terms of their attributes.
Using data on sales of farmland obtained through public auctions in England and
Wales, this paper examines the consequences of explicitly allowing for the presence of a spatio-temporal lag in the estimation of hedonic models of farmland
value. The results indicate that spatio-temporally lagged values of the dependent
and independent variables contribute significant additional explanatory power.
Accounting for spatio-temporal relationships appears moreover to somewhat alter
the perceived size and statistical significance of key farmland attributes.
Keywords: Farmland prices; hedonic analysis; spatial econometrics; spatio-temporal.
JEL classifications: C31, Q11, Q51.
1. Introduction
The hedonic (or Ricardian) technique has been widely used to value farmland amenities. It involves regressing variables related to the structural and environmental
characteristics of farmland on the rental value or sale price per unit area. Any differences in market values arising from differences in the abundance of key farmland
characteristics are attributed to the productive value of those characteristics when
the land is put to its most profitable use, given current input and output prices. The
theory underlying the hedonic model applied to agricultural land is well understood
(see, e.g. Palmquist, 1989 or Freeman, 1993 for an exposition).
Using this method, researchers have investigated the value of soil quality (Miranowski and Hammes, 1984); investments in drainage (Palmquist and Danielson,
1
David Maddison is based in the Department of Economics, University of Birmingham,
Edgbaston, Birmingham B15 2TT, UK. Tel.: +44 (0)121 414 6653, Fax: +44 (0)121 414
7377, E-mail: d.j.maddison@bham.ac.uk
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David Maddison
1989); top-soil depth (Brown and Barrow, 1985); measures to stem soil erosion
(Ervin and Mill, 1985); eligibility for government schemes (Nickerson and Lynch,
2001); the availability of water (Mendelsohn and Dinar, 2003); and the value of climate variables (Mendelsohn et al., 1994; Reinsborough, 2003).
Many of the farmland attributes valued by previous researchers using the hedonic
technique are spatially autocorrelated and vary significantly only over large geographical distances. Farmland attributes not observed by the researcher but nevertheless important to farmland productivity might also vary comparatively little over
short distances. Examples of important non-observed characteristics include the
nebulous but potentially important ‘distance to market’ and ‘land development
potential’. Currently, such factors are often controlled for by distance to the nearest
city or by the population density of the county in which the farmland is located,
although neither of these controls is adequate given the arbitrariness of deciding
what constitutes a city, much less a market.2
Individual farmers or institutional investors contemplating land purchases might
also place considerable weight on the price information generated by earlier sales
and in particular by the sale of nearby plots. Although such behaviour is at odds
with the idea underlying the hedonic technique, namely that farmers have the perfect information and care only about the characteristics of the plot, such behaviour
might be sensible when the collection of information is a costly activity. According
to Can (1992), such behaviour takes the form of seeking ‘comparable’ sales in earlier time periods.3
Noting the potential bias in parameter estimates that arises when these and
related issues are ignored, Bell and Dalton (2007) point out that controlling for spatial effects in studies necessitates a range of analytical modifications to hedonic
analyses. A failure to account for spatial phenomenon might lead to a violation of
the assumptions required to establish the optimal properties of ordinary least
squares (OLS), properties upon which hedonic analyses frequently depend. (For evidence on the propensity of spatially autocorrelated residuals to exaggerate the significance levels for any spatially autocorrelated explanatory variables, see Kramer
and Donninger, 1987.)
Despite the relatively large number of papers valuing farmland characteristics,
few studies of farmland value have so far adopted an explicitly spatial perspective.4
Such techniques are nevertheless becoming more widespread in the analysis of residential land markets. Examples include those of Pace and Gilley (1997), Basu and
Thibodeau (1998), Tse (2002) and Kim et al. (2003). (For a general discussion of
the use of spatial statistics in real estate, see Pace et al., 1998a.) Benirschka and
Binkley (1994) explain the variation in county-wide agricultural land prices while
accounting for the fact that the residual values for observations drawn from neighbouring counties are likely to be autocorrelated. Vandeveer et al. (1998) use Geographical Information Systems to illustrate the presence of spatial autocorrelation
2
See Dubin (1992) for a further discussion of the concept of ‘sliding neighbourhoods’. See
also Clonts (1970).
3
By ‘comparable’ I mean comparable in terms of plot characteristics.
4
Although trade in agricultural produce should in theory equalise the returns to farmland of
a given quality, some researchers have found it necessary to segment the land market into a
series of sub-markets in order to account for spatial heterogeneity (e.g. Elad et al., 1994).
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A Spatio-temporal Model of Farmland Values
173
in agricultural land prices. This visual evidence is reinforced by formal tests of spatial autocorrelation. Patton and McErlean (2003) found evidence of spatial relationships in their analysis of agricultural land prices in Northern Ireland. They allege
that agricultural land prices are characterised by a simultaneous spatial lag, which
they subsequently attribute to individuals relying on the price of adjacent plots of
land as a guide.5,6 In their study of the disamenities caused by large-scale animal
operations and mushroom production, Ready and Abdalla (2005) account for spatial autocorrelation in their hedonic house price study.
The contribution of this paper is to extend the hedonic model of farmland values
to one in which spatio-temporal relationships are present. More specifically, the
model presented here assumes that the price of farmland is affected by a spatially
weighted average of sale prices achieved during past time periods, arguing that Patton and McErlean’s implicit assumption of a simultaneous spatial lag is an implausible representation of the market price formation process. Individuals
contemplating the purchase of farmland can use only the past as a guide, not the
future. The model presented in this paper further allows for the possibility that the
characteristics of spatio-temporally lagged sales might be important in determining
the extent to which they serve as comparables. In making these changes, the paper
pays much closer attention than did previous researchers to the manner in which
land markets appear to function.
Anticipating the main results of the paper, adding spatio-temporally lagged values
of farmland values as well as terms describing the characteristics of those sales
generates a highly significant improvement in explanatory power. At the same time,
that statistical significance of key farmland attributes appears to change. Finally,
calculating the value of farmland amenities is far easier than in a model characterised by a simultaneous spatial lag. Such findings are of great concern to those seeking to use hedonic analysis to measure the value of farmland attributes accurately.
It is important to note that although it has a temporal component, the purpose of
this paper is not to explain the evolution of farmland prices over time. That is the purpose of models methodologically quite different in outlook to the one presented here
in which dummy variables are used to absorb all macroeconomic influences. The focus
is purely on hedonic models and how previously unexplained cross-sectional variation
in farmland values can be reduced and the risk of potential bias lessened by including
spatio-temporally lagged values of the dependent and independent variables.7
The remainder of the paper is organised as follows. Section 2 offers a suitably
brief review of the relevant aspects of spatial econometrics while at the same time
directing the reader to more detailed references in the literature. Section 3 describes
the dataset upon which the model is estimated. Section 4 presents a series of
5
See also Maddison (2004) and Patton and McErlean (2004).
There are also a number of spatial analyses or discussions relating to the applicability of
spatial techniques to agriculture but not involving farmland values. Examples include Bockstael (1996), Weiss (1996), Nelson and Hellerstein (1997), Irwin and Bockstael (2002) and Roe
et al. (2002). Holloway et al. (2007) provide an excellent survey of the literature. Florax and
Van der Vlist (2003) give an overview of the more general application of spatial econometric
techniques.
7
Numerous papers have analysed the determinants of aggregate land prices using time series
data (e.g. Lloyd et al., 1991).
6
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David Maddison
econometric models and section 5 presents a new spatio-temporal model of farmland values, which is then formally compared with a model characterised by a
simultaneous spatial lag. The final section concludes.
2. Spatial Econometrics
Given the several recent reviews of the spatial econometrics literature, the review
provided here will be brief.8
Spatial data may present itself in a variety of forms. Data can be observed at
irregularly spaced points on a two-dimensional surface. Alternatively, spatial data
can refer to geographical areas, some of which may be contiguous to other units.
Spatial relationships in regression analyses involving such data can also be modelled in a variety of ways. Spatial lags hypothesise that the value of the dependent variable observed at a particular location is partially determined by a spatially weighted
average of the value of the dependent variable as measured at other locations. Such a
model cannot be estimated by OLS because of the problem of simultaneity bias and
must instead be dealt with using either instrumental variable estimators or maximum
likelihood techniques. The spatial lag model in matrix form is given by:
Y ¼ a þ Xb þ qWY þ e
ð1Þ
whereY is a (n · 1) vector of dependent variables, X a (n · k) matrix of explanatory
variables, W is the (n · n) spatial weight matrix, a and q are scalar parameters and
b is a (k · 1) vector of parameters.
A number of different assumptions can be made concerning the spatial weights
matrix required to compute the spatially weighted values at each location. One possibility is that the weights matrix should merely identify the nearest neighbour or n
nearest neighbours. An alternative is that the weights matrix should identify those
observations located within an arbitrarily fixed distance. A third alternative is that
the weights matrix contains the inverse geographical distances between observations
or some function thereof. In instances in which data refer to geographical areas as
opposed to points on a plane it is customary to construct a contiguity matrix indicating which geographical areas are adjacent to one another. Weights matrices are
commonly row-standardised such that their rows sum to unity and have zeros along
the leading diagonal.
Calculating the impact of marginal changes in the value of the independent variables in the spatial lag model is not as straightforward as it might first appear. The
reason is that changes in the independent variables have both direct and indirect
feedback effects on the value of the dependent variable. This fact can best be appreciated by solving the spatial lag model for Y to obtain:
Y ¼ ðI qWÞ1 Xb þ ðI qWÞ1 e
ð2Þ
This is an important point to which I shall return.
8
For a straightforward introduction to spatial econometrics, see LeSage (1997) and Dubin
(1998). Detailed texts include Cliff and Ord (1981), Upton and Fingleton (1985), Anselin
(1988) and Cressie (1993). A collection of interesting papers can be found in Anselin and Florax (1995). Anselin (2002) and Florax and Van der Vlist (2003) provide a recent overview of
the literature. An important recent contribution can be found in Anselin (2003).
2008 The Author. Journal compilation 2008 The Agricultural Economics Society.
A Spatio-temporal Model of Farmland Values
175
An alternative form of spatial relationship occurs when the dependent variable
can also be predicted as a function of spatially lagged values of the independent
variables. Unlike spatial lags, this model can be estimated using OLS. The spatial
regression model is given by:
Y ¼ a þ Xb þ WXc þ e
ð3Þ
where c is a (k · 1) vector of coefficients.
The final way of incorporating spatial relationships is through spatial dependence
in the error term. This can be in the form of spatially autoregressive errors or a spatial moving average. The spatial error model is given by:
Y ¼ a þ Xb þ e
ð4Þ
e ¼ kWe þ u
ð5Þ
where
and k is a scalar parameter. Such models can be estimated through feasible generalised least squares or maximum likelihood techniques.
A distinction can be drawn between spatio-temporal and simultaneous spatial
relationships. In spatio-temporal models, the observations are ordered in time such
that spatially lagged values of current dependent and independent variables are
allowed to influence only the future realisations of random variables. For the purposes of this paper, the distinction between these two types of model is very important. Calculating the marginal impact of changes in the independent variables is
also much easier in the context of a spatio-temporal model.9 (For an introduction
to spatio-temporal models, see Pace et al., 1998b.)
Spatial relationships in the data typically violate the assumptions underlying
OLS, leading either to inefficiency and invalid hypothesis testing procedures or even
to bias and inconsistency in the parameter estimates. The specification of spatial
models can proceed either on the basis of diagnostic tests or on the basis of theoretical reasoning. Another important consideration is that simultaneously modelling
several forms of spatial relationships such as the spatial lags and a spatially autoregressive error term is not possible at present. A final consideration is that whereas
several tests are available that indicate whether spatial relationships are present in
data, some of these tests have difficulty in discriminating between different forms of
spatial relationship.
3. Data
The dataset used to test for the presence of spatio-temporal relationships in agricultural land prices and to illustrate the consequences of failing to account for such
relationships comprises observations on farmland transactions that occurred in England and Wales from the beginning of January 1994 to the end of June 1996. These
are taken from the twice-yearly publication Farmland Market. In total, 601 observations are available for analysis although in order to generate a spatio-temporal lag
(see below) the first six months worth of data (94 observations) are kept in reserve.
9
This is of obvious concern to anyone wishing to calculate the implicit price of farmland
characteristics in hedonic price models. Maddison (2004) provides more details.
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David Maddison
A shortcoming of the dataset is that only those sales taking place through public
auction are included in the analysis. But as a record of farms sold through public
auction, the dataset is almost complete.10 Moreover, the sale of land through
public auction rather than private transaction generates precisely the kind of information externality that was earlier argued to give rise to spatio-temporal relations
in the data.
The dataset includes the current sale price of the farmland (PRICE) measured in
pounds sterling and also the size of the farm (ACRES) in acres. Many sales have
large farmhouses attached to them. Usually information is available only on the
number of bedrooms, and the presence of other agricultural buildings is not
recorded systematically. The number of bedrooms is divided by the number of acres
(BEDS ⁄ ACRES) to provide an indicator of the level of structural attributes per unit
of land. Animals and agricultural equipment are typically sold separately from the
land and its buildings. The observations are geographically referenced using the Ordnance Survey gazetteer (a database of all place names in Britain and their spatial
coordinates). Observations, whose location could not be determined to a satisfactory
degree of precision, were discarded. Repeat sales of land were also discarded.
Consistent with previous studies, a variable is included denoting the distance in
kilometres to the nearest city, partly as a measure of distance to market and partly
as a measure of development potential. A city is variously defined with at least
250,000 people to obtain the variable DIST250 or with at least 100,000 people to
obtain the variable DIST100. Note that there are 17 cities with a population size in
excess of 250,000 in Great Britain and 57 cities with a population size in excess of
100,000. A referee remarks that proximity to smaller urban areas might also be
important. The problem about including a variable indicating proximity to urban
areas with a population of significantly less than 100,000 is that the variation in the
resulting variable is sharply diminished. Observe already the difference in the standard deviation between variables DIST250 and DIST100 in Table 1. Alternately
put, nowhere in England or Wales is very far from a community of 10,000 or more.
Following Shi et al. (1997), I also create a variable URBAN which takes into
Table 1
The dataset
Variable
PRICE ⁄ ACRES
ACRES
BEDS
ALC
DIST250
DIST100
URBAN
Mean
SD
Minimum
Maximum
2,814
84
1.14
3.00
57
36
5,867
1,441
148
2.06
0.82
38
20
2
154
7
0
1
5
2
5,866
12,571
2,794
11
5
211
123
5,900
10
Even if this dataset were extended to include farms sold by private treaty, it would, of
course, still represent only a small fraction of the farmland in England and Wales. This is a
problem of most hedonic analyses, although see Koundouri and Pahshardes (2003) for an
exception.
2008 The Author. Journal compilation 2008 The Agricultural Economics Society.
A Spatio-temporal Model of Farmland Values
177
account both geographical distances and the population size of nearby cities. This
involves calculating for each plot i and for each city j of at least 100,000 people the
following variable:
X
URBANi ¼
POPj =DIST2ij
ð6Þ
j
where POP is the population of city j · 100,000 and DIST is the distance between
plot i and city j in kilometres. Including these variables does not of course preclude
the possibility that the spatial techniques employed below will prove better at modelling the twin influences of distance to market and development potential.
A series of land-quality variables were matched to the data using the geographical
coordinates of the farmland sold. The 5-km grid square agricultural land classification (ALC) system of England and Wales classifies land into one of five different
grades (Ministry of Agriculture Food and Fisheries, 1988). The land-grading system
is based on the extent to which among other things, climate, physical composition,
drainage characteristics and slope impose long-term limitations on lands’ agricultural use. Grade 1 agricultural lands are the best and grade 5 the worst. The same
dataset also records if land was non-agricultural in nature or simply not surveyed in
which case it is dropped from the dataset.
The obvious disadvantage of the ALC is that it fuses together a number of characteristics, making it impossible to determine the individual contribution any given
characteristic makes to farmland productivity. The advantage of using the classification is that it is the result of many years of fieldwork and is available at a high level
of spatial resolution. As using data at a low level of spatial resolution can itself
result in spatial autocorrelation and because the focus of this paper is primarily
methodological, on balance it was deemed appropriate to use the classification as
the sole measure of farmland productivity.
The most natural way of incorporating the land quality data is obviously to
include four dummy variables corresponding to the different land classifications,
along with the constant term. But in what follows, a single variable taking integer
values from 1 to 5 was included in the model. This is because the majority of land
is classified as either grade 3 or 4 and because including separate dummies for each
land classification did not result in a statistically significant improvement in fit in
any of the regression models presented below.
Finally, dummy variables DUM1, DUM2 and DUM3 identify the periods January 1995 to June 1995, July 1995 to December 1995 and January 1996 to June
1996, respectively. The constant term refers to the period July 1994 to December
1994. The purpose of these dummy variables is to account for changes in the current price of inputs and outputs as well as future expectations regarding these
things. Changes in price expectations are clearly very important in determining the
value of land and failing to control for these risks obscuring the value of farmland
attributes. The dataset is described in Table 1.
The Euclidean distance between each parcel of land is computed and used to construct a distance matrix D. The median distance between farmland sold by auction
during the period July 1994 to June 1996 is 182 km. The minimum distance is 1 km
and the maximum distance is 644 km. The largest minimum distance is 74 km (i.e.
one farm is 74 km from its nearest neighbour) and the smallest maximum distance
is 329 km (i.e. one centrally located farm is within 329 km of every farm in the
dataset). In order to construct the weights matrix, W, required for the ensuing
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David Maddison
calculations, it is convenient to assume that the individual elements (wij) of the
weight matrix evolve according to:
f
wij ¼ 1=d ij
ð7Þ
where the exponent f is often referred to as the friction parameter and dij are the
elements of D.11 The leading diagonal of the weights matrix W is filled with zeros
and the matrix is then row-standardised such that the rows of the matrix each sum
to unity. An alternative assumption is to compute a standardised binary spatial
weight matrix identifying those farms which lie within a pre-specified distance of
one another. Unfortunately, the largest minimum distance between sales contained
in the dataset (74 km) limits the usefulness of this approach. But by increasing the
size of the friction parameter, the weight placed on more distant farms can be made
arbitrarily small and does away with the need to specify a distance limit. I have
examined different values for f and find that the inverse distance model with f = 1
yields the most pronounced evidence of spatial dependence.
4. Econometric Results
The presence of spatial autocorrelation in the data is confirmed using Geary’s
C-statistic (Geary, 1954). The results of these tests are presented in Table 2 and
indicate that as might be anticipated, the variables PRICE ⁄ ACRES and ALC are
significantly autocorrelated over space. The level of structural attributes per unit of
land BEDS ⁄ ACRES is spatially autocorrelated at the 5% level of confidence and
ACRES is not spatially autocorrelated. The fact that some explanatory variables
are highly spatially autocorrelated, whereas others are not, suggests that accounting
for spatial effects might have a differential impact on the perceived size and statistical significance of these variables.
Using an alternative approach to examine the range of spatial autocorrelation, a
series of binary standardised spatial weight matrices were created identifying neighbouring farms that fall within, e.g. 0–75, 75–150, 150–225 km of one other. Computing Geary’s C-statistic for each of these binary spatial weight matrices suggests
that the price per acre of farms within 150 km of each other exhibits statistically
significant positive spatial autocorrelation and that for farms within 75 km of each
other this statistical relationship is highly significant. This same test is also conducted using cumulated distance, i.e. for farms that fall within 0–75, 0–150,
Table 2
Global spatial autocorrelation statistics
Variable
PRICE ⁄ ACRES
ALC
BEDS
ACRES
Geary’s C
Probability
0.939
0.846
0.978
1.042
0.000
0.000
0.010
0.253
Note: Spatial weight matrix = inverse distance row-standardised.
11
See, for example, the STATA Technical Bulletin STB-60 published in March 2001.
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A Spatio-temporal Model of Farmland Values
Table 3
Banded measures of spatial correlation
Distance (km)
0–75
75–150
150–225
225–300
300–375
375–450
450–525
525–600
Geary’s C
)5.493
)1.828
)0.275
1.298
)0.078
)0.650
)0.657
)0.534
(Prob
(Prob
(Prob
(Prob
(Prob
(Prob
(Prob
(Prob
=
=
=
=
=
=
=
=
Distance (km)
0.000)
0.034)
0.392)
0.097)
0.469)
0.258)
0.256)
0.297)
0–75
0–150
0–225
0–300
0–375
0–450
0–525
0–600
Geary’s C
)5.493
)3.940
)1.815
0.020
1.276
1.095
0.903
0.729
(Prob
(Prob
(Prob
(Prob
(Prob
(Prob
(Prob
(Prob
=
=
=
=
=
=
=
=
0.000)
0.000)
0.035)
0.492)
0.101)
137)
0.183)
0.233)
Notes: Spatial weight matrix = inverse distance row-standardised.
Variable = PRICE ⁄ ACRE.
0–225 km of one another. This suggests spatial autocorrelation even up to a range of
225 km. Note that the probability values given in Table 3 refer to a one-tailed test.
To begin with, various special cases of the following model are estimated:
log(PRICE/ACRES) ¼ a þ b1 ACRES þ b2 ALC þ b3 (BEDS/ACRES)
þ b4 (BEDS/ACRES)2 þ b5 DIST250 þ b6 DIST100
þ b7 URBAN þ b8 DUM1 þ b9 DUM2
X
þ b10 DUM3 þ q
wi log(PRICE/ACRES)i þ e
where:
e¼k
X
w i ei þ u
ð8Þ
ð9Þ
and a, b1–10, q and k are the parameters. Note that the dependent variable was
taken as the log of PRICE ⁄ ACRES following attempts to estimate a Box–Cox
transformation of the dependent variable. Taking PRICE ⁄ ACRES as a purely linear variable resulted in an equation which did not pass the RESET test for functional form (Ramsey, 1969). For the same reason, it was necessary to add a
quadratic term for BEDS ⁄ ACRES. The number of acres was included as an explanatory variable in case repackaging of land is a costly activity.12
Setting the parameters q and k equal to zero yields the usual hedonic price equation estimable by OLS. Allowing the parameter q to vary results in a model in
which individual sale prices are affected by the prices of neighbouring properties.
Allowing the parameter k to vary results in a model where the residuals are autocorrelated over space.13
The regression results are presented in Table 4 in which the conventional hedonic
model estimated by OLS with q and k are set equal to zero. Note that the t-statistics
12
The so-called ‘linearity hypothesis’ (Parsons, 1990) would imply taking the dependent variable PRICE in levels and multiplying any environmental attributes like ALC by the number
of acres. Here I prefer to take the dependent variable as price per acre in order to reduce heteroscedasticity so I divide the structural attribute BEDS by the number of acres.
13
All calculations are undertaken using STATA.
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David Maddison
Table 4
The regression results for the OLS model
Variable
CONSTANT
ACRES
ALC
BEDS ⁄ ACRES
(BEDS ⁄ ACRES)2
DIST250
DIST100
URBAN
DUM1
DUM2
DUM3
F-statistic
No. obs.
R2
RESET
Parameter (t-stat)
61.819
)0.000793
)0.132
11.359
)20.609
)0.00281
)0.00142
)0.00913
0.2183
0.295
0.355
87.74
507
0.407
0.05
(1.81)
()6.31)
()5.98)
(13.84)
()10.24)
()3.80)
()1.28)
()1.57)
(4.64)
(5.83)
(6.99)
(Prob=0.000)
Parameter (t-stat)
8.165
)0.000794
)0.137
11.288
)20.591
)0.00303
(Prob=0.983)
0.220
0.297
0.352
103.38
507
0.405
0.11
(110.69)
()6.34)
()6.34)
(13.80)
()10.28)
()4.89)
(4.68)
(5.87)
(6.97)
(Prob=0.000)
(Prob=0.951)
Notes: Dependent variable = log(PRICE ⁄ ACRES).
Method = OLS.
t-statistics are heteroscedastic-consistent.
are based upon robust standard errors. The results reveal that as land quality worsens the price of land declines. The results also indicate that the greater the number
of bedrooms per acre the greater the price albeit at a diminishing rate. Smaller plot
sizes command a higher price per acre indicating that the repackaging of land is a
costly activity. Turning now to the alternative measures of distance to market cum
urban development potential, defining a city in terms of an urban area with a
population in excess of 250,000 is by far the most successful strategy in terms of fit.
This variable is negatively signed as anticipated, and highly significant. By contrast,
DIST100 and URBAN are jointly insignificant and dropped from all further
regressions.14
Before conducting any spatial regressions, an important issue to address is
whether the regressions are stable across different regions or whether geographically
distinct markets exist, within which the implicit prices of farmland characteristics
differ (Straszheim, 1974). Freeman (1993) explains the circumstance under which
geographically segmented markets arise: when buyers do not, for whatever reason,
participate in different markets while at the same time these regions differ in terms
of the supplies of and demands for attributes.
When markets are segmented but model intercepts and coefficients are constrained to be the same, market segmentation can manifest itself in positive tests for
spatial autocorrelation. There is however no prior information to indicate the areas
that might potentially be considered separate markets, and furthermore the sample
size is of modest dimensions. These considerations suggest that the most appropriate strategy is to use the dummy variable approach to allow the intercept and slopes
of the regression equations to differ across each of four quadrants referring to the
14
F(2, 496) = 1.57.
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A Spatio-temporal Model of Farmland Values
Table 5
The regression results for the OLS model with regional land markets
Variable
SE parameter
(t-stat)
NE parameter
(t-stat)
NW parameter
(t-stat)
CONSTANT
7.708 (43.81)
8.0380 (55.88)
8.690
ACRES
0.000117 (0.31) )0.000901 ()5.55) )0.000678
ALC
)0.0281 ()0.58)
)0.116 ()2.86)
)0.259
BEDS ⁄ ACRES
11.285 (3.56)
11.621 (9.13)
10.960
(BEDS ⁄ ACRES)2 )6.212 ()0.22)
)22.526 ()7.47)
)23.920
DIST250
0.000127 (0.09)
0.000004 (0.00)
)0.00343
DUM1
0.0901 (1.14)
0.269 (3.17)
0.247
DUM2
0.167 (2.21)
0.149 (1.55)
0.278
DUM3
0.147 (1.71)
0.380 (4.16)
0.428
F-statistic
50.52 (Prob=0.000)
No. obs.
507
R2
0.486
RESET
1.34 (Prob=0.260)
SW parameter
(t-stat)
(39.20)
7.911 (55.12)
()2.27) )0.000944 ()1.47)
()4.03)
)0.0902 ()2.41)
(2.89)
7.172 (2.00)
()0.55)
24.868 (0.78)
()3.12) )0.00321 ()2.93)
(2.08)
0.1403 (1.66)
(2.06)
0.503 (5.41)
(3.59)
0.424 (4.51)
Notes: Dependent variable = log(PRICE ⁄ ACRES).
Method = OLS.
t-statistics are heteroscedastic-consistent.
south-east (SE), the north-east (NE), the north-west (NW) and the south-west
(SW). The coefficients relating to these geographically defined sub-markets are presented in Table 5. The test for parameter stability is significant at the 1% level of
confidence, implying that the hypothesis of a market for farmland, unified by trade
in agricultural commodities, must be abandoned.15 Across the four sub-markets,
several important differences are observed in the implicit values of farmland characteristics. In the SE and the SW, the price per acre is not affected by farm size. In
the SE, high-quality farmland does not attract a premium possibly because it is relatively more abundant there. Alternatively, the value of agricultural land grade 3, 4
and 5 may be higher in the SE because of the greater developmental opportunities
typically denied to prime agricultural land or because of the higher recreational and
environmental values in part related to the higher population and greater wealth of
the SE. Finally, distance to the nearest big city is an amenity only for farms in the
NW and SW of the country.
Table 6 displays test results for spatial error and spatial lag dependence in the
OLS model with heterogeneous land markets. The Lagrange multiplier tests are less
useful in this respect because each test has some power against the alternative form
of spatial dependence. The robust Lagrange multiplier tests overcome this problem
(Anselin et al., 1996). Both the robust and the non-robust versions of the test statistics point to the existence of a spatial lag.16 There is no evidence that spatial autocorrelation is due to omitted neighbourhood characteristics (the tests for spatial
error dependence are statistically insignificant).
15
F(27, 471) = 3.92.
Note that there are other possible causes of a significant spatial correlation, e.g. use of an
incorrect functional form.
16
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David Maddison
Table 6
Tests of spatial autocorrelation in the OLS model
Spatial error dependence
Moran’s I-statistic
Lagrange multiplier test
Robust Lagrange multiplier test
Spatial lag dependence
Lagrange multiplier test
Robust Lagrange multiplier test
3.631 (Prob=0.000)
2.191 (Prob=0.139)
2.910 (Prob=0.088)
9.134 (Prob=0.003)
9.852 (Prob=0.002)
Note: Spatial weight matrix = inverse distance row-standardised.
Table 7
The regression results for the spatial lag model
Variable
SE parameter
(T-stat)
NE parameter
(T-stat)
NW parameter
(T-stat)
CONSTANT
2.013 (1.21)
2.320 (1.39)
2.866
ACRES
0.000102 (0.28) )0.000910 ()5.65) )0.000635
ALC
)0.0237 ()0.50)
)0.108 ()2.81)
)0.252
BEDS ⁄ ACRES
11.172 (3.53)
11.253 (9.23)
10.833
(BEDS ⁄ ACRES)2
)2.724 ()0.10) )21.0168 ()7.19) )21.969
DIST250
0.000440 (0.31)
0.000313 (0.31)
)0.00206
DUM1
0.0983 (1.28)
0.252 (3.13)
0.243
DUM2
0.180 (2.43)
0.144 (1.56)
0.286
DUM3
0.164 (1.93)
0.367 (4.31)
0.410
q
0.721 (3.43)
Log-likelihood
)198.0011
No. obs.
507
Square correlation
0.498
(pseudo-R2)
SW parameter
(T-stat)
(1.68)
2.239 (1.35)
()2.31) )0.00102 ()1.72)
()4.17) )0.0772 ()2.13)
(3.03)
6.113 (1.83)
()0.53)
34.252 (1.18)
()1.89) )0.00263 ()2.45)
(2.18)
0.150 (1.90)
(2.27)
0.286 (2.27)
(3.64)
0.425 (4.72)
Notes: Dependent variable = log(PRICE ⁄ ACRES).
Method = ML (Maximum Likelihood).
Spatial weight matrix = inverse distance row-standardised.
t-statistics are heteroscedastic-consistent.
Table 7 presents the same regression equation but with spatial lag dependence
(q allowed to vary but k is set equal to zero). The parameter estimates are not much
altered but the spatial autocorrelation parameter q is highly significant.17 The
results from this model are very similar to those from the heterogeneous land market model without a spatial lag except that distance to the nearest big city is no
longer significant in the NW whereas previously it was significant at the 1% level of
confidence. But it is important to avoid believing that because the parameter coefficients are similar that the implicit value of marginal changes in the value of the
dependent variables must also be similar. The preceding section illustrated that
17
Note that I do not allow this parameter to vary over sub-markets in the specification
shown in Table 7.
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A Spatio-temporal Model of Farmland Values
183
calculating the marginal value of changes in the dependent variable in a model characterised by a spatial lag is quite involved.
5. A Spatio-temporal Model
The chief limitation of the spatial lag model presented in Table 7 which has so far
characterised the literature is that it ignores entirely the temporal nature of the
data at the expense of focusing on the spatial aspects. But as hedonic datasets are
almost invariably composed of a sequence of sales, the assumption of a simultaneous spatial lag present in earlier analyses is an impossible representation of the
price formation process. In effect, it suggests that the price of a plot of farmland
is determined by the price of plots as yet unsold. More plausible is to assume that
farmland prices depend on the price of recently sold farmland and, in particular,
on the price of recently sold farmland in the immediate vicinity.18 This suggests
including the spatio-temporally weighted values of the dependent variable as an
explanatory variable and this in turn requires the creation of a spatio-temporal
weight matrix.
The role of the spatio-temporal weight matrix is to identify inverse distancebased standardised weights for each observation based upon a temporally defined
subset of the observations. This matrix Z is obtained from the Hadamard product
of matrices W (previously defined) and T. If sales are ordered by time, then matrix
T is lower triangular and comprises a null matrix interspersed with blocks of ones
identifying for each observation on farmland sold in an earlier time period. The
construction of this matrix also involves making a decision regarding the length of
the time period over which individuals base their information. Because the dataset
here is of modest dimensions, it is arbitrarily assumed that farmland values are
determined by spatially weighted sale prices realised in the preceding six months.19
Finally, the matrix Z is once more row-standardised. Most of the computational
effort involved in spatio-temporal analysis goes into the construction of this
matrix.
A second limitation of the spatial lag model is that it assumes that the price
of farmland is affected by the price per acre of nearby land but not by the characteristics of the land that was sold. In such a model, individuals risk basing
their judgements on land sales that are not comparable in terms of characteristics.20 This deficiency is easily remedied by including the spatio-temporally lagged
values of farmland characteristics as additional explanatory variables. Including
18
There is one way of rescuing the simultaneous spatial lag model. That is, if the same market participants are involved in the bidding. The fact that we use data observed at six-month
intervals would seem to diminish this possibility. The simultaneous spatial lag model might
also be appropriate if farmers were asked about their perceptions of the worth of their land
at a particular point in time.
19
Without a far longer time period, it is not possible to answer the question: How far back
in time do buyers make comparisons? Presumably, sales in the recent past carry most weight.
20
The information conveyed by the fact that someone was prepared to pay £10,000 per acre
for a plot of nearby land is for example quite different depending on whether this land was
grade 1 agricultural land with a five-bedroom farmhouse or a plot with grade 4 agricultural
lands and no dwelling.
2008 The Author. Journal compilation 2008 The Agricultural Economics Society.
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David Maddison
the spatio-temporally lagged values of the dependent and independent variables
as advocated by Pace et al. (1998b) and Can and Megbolugbe (1997) yields the
following equation:
log(PRICE/ACRES) ¼ a þ b1 ACRES þ b2 ALC þ b3 (BEDS/ACRES)
þ b4 (BEDS/ACRES)2 þ b5 DIST250 þ b6 DUM1
X
zi log(PRICE/ACRES)i
þ b7 DUM2 þ b8 DUM3 þ q
X
X
þ c1
zi ACRESi þ c2
zi ALCi
X
X
þ c3
zi (BEDS/ACRES)i þ c4
zi (BEDS/ACRES)2i
X
þ c5
zi DIST250i þ e:
ð10Þ
This model is estimated by OLS jointly for each sub-market and, from the results
in Table 8, it emerges that the spatio-temporally lagged variables are significant as a
group at the 1% level of confidence.21,22 Furthermore, there is no evidence of any
outstanding spatial autocorrelation in the residuals in the regression. Moreover,
compared with the model in Table 7 the squared correlation (or R2) statistic has
risen from 0.49 to 0.52.
Apart from the greater inherent plausibility of the spatio-temporal model and the
evident importance of accounting for spatio-temporally lagged values of the dependent variables there are two other advantages of the spatio-temporal model. First,
unlike the spatial lag model, it is easy to calculate the implicit value of marginal
changes in the level of the characteristics as there is no simultaneity present.23 Second, the model can be estimated using OLS.
The main reason for being concerned about spatial autocorrelation in hedonic
datasets is to restore the optimal properties of the estimation techniques. This has
been achieved by including a set of spatio-temporal variables whose contribution to
the explanatory power of the regression equation was found to be highly significant
(Lagrange multiplier tests reveal no trace of spatial autocorrelation). It remains only
to compare the parameter estimates from the spatio-temporal model with those
taken from the heterogeneous market model presented in Table 5 in order to determine what if anything is changed.
The coefficient on ACRES in the NW quadrant is now insignificant at the 5%
level of confidence. The coefficients on ALC remain statistically significant at the
5% level of confidence for the NE, NW and SW. The coefficient on BEDS ⁄ ACRES
becomes statistically insignificant for the SW. Of particular interest is the fact that
distance to the nearest large city, which was highly significant in both the NW and
SW is now no longer significant. The correct interpretation is presumably not that
21
Because of the absence of simultaneity, such models can be consistently estimated by OLS.
F(24, 447) = 2.95.
23
For example, Table 8 immediately indicates that there is a 10.5% difference in land values
between grade 2 and grade 3 agricultural lands in the NE, and a 24.9% difference in the
NW. The inclusion of spatio-temporal terms does not complicate the calculation of implicit
prices and serves only to eliminate any potential bias that might otherwise arise. For a discussion relating to the calculation of the implicit value of amenities in the spatial lag model,
see Maddison (2004).
22
2008 The Author. Journal compilation 2008 The Agricultural Economics Society.
The regression results for the spatio-temporal model
Variable
CONSTANT
ACRES
ALC
BEDS ⁄ ACRES
(BEDS ⁄ ACRES)2
DIST250
DUM1
DUM2
DUM3
P
Pzi · log(PRICE ⁄ ACRE)i
Pzi · ACRESi
Pzi · ALCi
Pzi · BEDS ⁄ ACRESi 2
Pzi · (BEDS ⁄ ACRES) i
zi · DIST250i
F-statistic
No. obs.
R2
RESET
SE parameter (T-stat)
)8.546
)0.000162
)0.0340
11.715
)6.306
0.00181
0.689
)0.377
)0.516
2.120
)0.000271
0.140
66.556
)1274.378
)0.00541
()1.31)
()0.43)
()0.79)
(4.16)
()0.23)
(0.91)
(2.58)
()2.24)
()2.05)
(2.40)
()0.07)
(0.67)
(1.51)
()2.23)
()0.54)
NE parameter (T-stat)
12.631
)0.000920
)0.105
11.167
)20.857
0.00241
0.421
0.396
0.473
)0.303
)0.00562
)0.354
19.784
)153.970
)0.0221
NW parameter (T-stat)
(2.09)
7.974
()6.15)
)0.000452
()2.19)
)0.249
(7.54)
9.999
()5.62)
)2.482
(1.74)
0.00118
(3.56)
0.2411
(1.71)
0.228
(2.32)
0.150
()0.41)
0.354
()1.62)
)0.00150
()1.41)
)0.562
(1.00)
28.0051
()1.12)
)36.00454
()2.44)
)0.0169
36.88 (Prob=0.000)
507
0.523
1.68 (Prob=0.171)
(1.22)
()1.67)
()3.39)
(2.66)
()0.05)
(0.46)
(1.55)
(1.01)
(0.64)
(0.51)
()1.48)
()0.83)
(2.32)
()0.46)
()1.75)
SW parameter (T-stat)
7.489
)0.000982
)0.0803
5.474
45.0216
)0.00338
0.0787
0.482
0.383
0.202
0.00323
)0.428
)24.343
250.238
0.000870
(2.32)
()1.46)
()1.98)
(1.49)
(1.38)
()1.49)
(0.68)
(3.51)
(1.54)
(0.50)
(0.97)
()1.88)
()0.92)
(0.84)
(0.17)
A Spatio-temporal Model of Farmland Values
2008 The Author. Journal compilation 2008 The Agricultural Economics Society.
Table 8
Notes: Dependent variable = log(PRICE ⁄ ACRES).
Spatio-temporal weight matrix = inverse distance row-standardised.
t-statistics are heteroscedastic-consistent.
185
186
David Maddison
distance to market or development potential are unimportant, but rather that
including spatio-temporally lagged variables does a better job of absorbing the variation in the data caused by proximity to the market and development potential than
distance to the nearest big city. Such findings draw attention to the importance of
the spatio-temporal modelling techniques outlined in this paper.
Despite the fact that they are highly significant as a group, the coefficients on the
spatio-temporal variables have relatively large standard errors. The reason is that
the spatio-temporal lags in the dependent and independent variables are somewhat
correlated with each other. The coefficients on the spatio-temporal terms are not,
however, the focus of attention in a hedonic regression. I attempted to simplify the
model by excluding the spatio-temporally lagged values of the independent variables, but this restriction is rejected at the 1% level.24 This highlights the importance of controlling for comparability.
Spatio-temporal models can be extended in a number of directions. It is in principle possible to increase the number of spatio-temporal lags to include sales from
earlier time periods which is not done here because of the relatively short time period covered by the dataset. In the same way that buyers apparently take relatively
greater note of nearby properties, it seems probable that they also take relatively
greater note of more recent sales.
6. Conclusions
This paper has presented a spatio-temporal model in which farmland values are
affected not only by the price per acre of nearby land sold in the preceding six
months but also by the characteristics of the land that was sold. This paper builds
on earlier research indicating that agricultural land prices might be characterised by
a spatial lag. Compared with the spatial lag model, the formulation in this paper
appears to possess a number of advantages. First it acknowledges the role of comparables in determining prices. Second, it can be estimated using OLS and third it
makes it easier to calculate the implicit price of farmland characteristics as there is
no simultaneity present.
Accounting for spatio-temporal relationships also appears to have some impact
on the perceived value and statistical significance of the characteristics of farmland.
This is not surprising given the fact that some of the characteristics of farmland are
very highly spatially correlated. The spatio-temporal model also appears to do a
better job of controlling for hard to model but conceptually important local influences such as distance to market and development potential. The spatio-temporal
nature of farmland sale price data should not be ignored in empirical studies. Likewise, it is probable that spatio-temporal relationships are also important in other
hedonic applications.
Acknowledgements
The author would like to acknowledge the helpful comments of the editor and
two anonymous referees. Any errors are however the sole responsibility of the
author.
24
F(20, 447) = 2.15.
2008 The Author. Journal compilation 2008 The Agricultural Economics Society.
A Spatio-temporal Model of Farmland Values
187
References
Anselin, L. ‘Model validation in spatial econometrics: A review and evaluation of alternative
procedures’, International Regional Science Review, Vol. 11, (1988) pp. 279–316.
Anselin, L. ‘Under the hood issues in the specification and interpretation of spatial regression
models’, Agricultural Economics, Vol. 27, (2002) pp. 247–267.
Anselin, L. ‘Spatial externalities spatial multipliers and spatial econometrics’, International
Regional Science Review, Vol. 26, (2003) pp. 153–166.
Anselin, L. and Florax, R. New Directions in Spatial Econometrics (Berlin: Springer Verlag,
1995).
Anselin, L., Bera, K., Florax, R. and Yoon, M. ‘Simple diagnostic tests for spatial dependence’, Regional Science and Urban Economics, Vol. 26, (1996) pp. 77–104.
Basu, S. and Thibodeau, T. ‘Analysis of spatial autocorrelation in house prices’, Journal of
Real Estate Finance and Economics, Vol. 17, (1998) pp. 61–86.
Bell, K. and Dalton, T. ‘Spatial economic analysis in data-rich environments’, Journal of
Agricultural Economics, Vol. 58, (2007) pp. 487–501.
Benirschka, M. and Binkley, J. ‘Land price volatility in a geographically dispersed market’,
American Journal of Agricultural Economics, Vol. 76, (1994) pp. 185–195.
Bockstael, N. ‘Mo
doi: 10.1111/j.1477-9552.2008.00182.x
A Spatio-temporal Model of Farmland
Values
David Maddison1
(Original submitted November 2007, revision received August 2008, accepted
September 2008.)
Abstract
Despite the fact that data on farm sales are invariably collected over both time
and space, previous papers have allowed for the presence of either temporal or spatial relationships in the data, but not both. Some papers have also inadvertently
assumed that although farmland values are influenced by prices realised by nearby
sales, these sales need not necessarily be comparable in terms of their attributes.
Using data on sales of farmland obtained through public auctions in England and
Wales, this paper examines the consequences of explicitly allowing for the presence of a spatio-temporal lag in the estimation of hedonic models of farmland
value. The results indicate that spatio-temporally lagged values of the dependent
and independent variables contribute significant additional explanatory power.
Accounting for spatio-temporal relationships appears moreover to somewhat alter
the perceived size and statistical significance of key farmland attributes.
Keywords: Farmland prices; hedonic analysis; spatial econometrics; spatio-temporal.
JEL classifications: C31, Q11, Q51.
1. Introduction
The hedonic (or Ricardian) technique has been widely used to value farmland amenities. It involves regressing variables related to the structural and environmental
characteristics of farmland on the rental value or sale price per unit area. Any differences in market values arising from differences in the abundance of key farmland
characteristics are attributed to the productive value of those characteristics when
the land is put to its most profitable use, given current input and output prices. The
theory underlying the hedonic model applied to agricultural land is well understood
(see, e.g. Palmquist, 1989 or Freeman, 1993 for an exposition).
Using this method, researchers have investigated the value of soil quality (Miranowski and Hammes, 1984); investments in drainage (Palmquist and Danielson,
1
David Maddison is based in the Department of Economics, University of Birmingham,
Edgbaston, Birmingham B15 2TT, UK. Tel.: +44 (0)121 414 6653, Fax: +44 (0)121 414
7377, E-mail: d.j.maddison@bham.ac.uk
2008 The Author. Journal compilation 2008 The Agricultural Economics Society.
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David Maddison
1989); top-soil depth (Brown and Barrow, 1985); measures to stem soil erosion
(Ervin and Mill, 1985); eligibility for government schemes (Nickerson and Lynch,
2001); the availability of water (Mendelsohn and Dinar, 2003); and the value of climate variables (Mendelsohn et al., 1994; Reinsborough, 2003).
Many of the farmland attributes valued by previous researchers using the hedonic
technique are spatially autocorrelated and vary significantly only over large geographical distances. Farmland attributes not observed by the researcher but nevertheless important to farmland productivity might also vary comparatively little over
short distances. Examples of important non-observed characteristics include the
nebulous but potentially important ‘distance to market’ and ‘land development
potential’. Currently, such factors are often controlled for by distance to the nearest
city or by the population density of the county in which the farmland is located,
although neither of these controls is adequate given the arbitrariness of deciding
what constitutes a city, much less a market.2
Individual farmers or institutional investors contemplating land purchases might
also place considerable weight on the price information generated by earlier sales
and in particular by the sale of nearby plots. Although such behaviour is at odds
with the idea underlying the hedonic technique, namely that farmers have the perfect information and care only about the characteristics of the plot, such behaviour
might be sensible when the collection of information is a costly activity. According
to Can (1992), such behaviour takes the form of seeking ‘comparable’ sales in earlier time periods.3
Noting the potential bias in parameter estimates that arises when these and
related issues are ignored, Bell and Dalton (2007) point out that controlling for spatial effects in studies necessitates a range of analytical modifications to hedonic
analyses. A failure to account for spatial phenomenon might lead to a violation of
the assumptions required to establish the optimal properties of ordinary least
squares (OLS), properties upon which hedonic analyses frequently depend. (For evidence on the propensity of spatially autocorrelated residuals to exaggerate the significance levels for any spatially autocorrelated explanatory variables, see Kramer
and Donninger, 1987.)
Despite the relatively large number of papers valuing farmland characteristics,
few studies of farmland value have so far adopted an explicitly spatial perspective.4
Such techniques are nevertheless becoming more widespread in the analysis of residential land markets. Examples include those of Pace and Gilley (1997), Basu and
Thibodeau (1998), Tse (2002) and Kim et al. (2003). (For a general discussion of
the use of spatial statistics in real estate, see Pace et al., 1998a.) Benirschka and
Binkley (1994) explain the variation in county-wide agricultural land prices while
accounting for the fact that the residual values for observations drawn from neighbouring counties are likely to be autocorrelated. Vandeveer et al. (1998) use Geographical Information Systems to illustrate the presence of spatial autocorrelation
2
See Dubin (1992) for a further discussion of the concept of ‘sliding neighbourhoods’. See
also Clonts (1970).
3
By ‘comparable’ I mean comparable in terms of plot characteristics.
4
Although trade in agricultural produce should in theory equalise the returns to farmland of
a given quality, some researchers have found it necessary to segment the land market into a
series of sub-markets in order to account for spatial heterogeneity (e.g. Elad et al., 1994).
2008 The Author. Journal compilation 2008 The Agricultural Economics Society.
A Spatio-temporal Model of Farmland Values
173
in agricultural land prices. This visual evidence is reinforced by formal tests of spatial autocorrelation. Patton and McErlean (2003) found evidence of spatial relationships in their analysis of agricultural land prices in Northern Ireland. They allege
that agricultural land prices are characterised by a simultaneous spatial lag, which
they subsequently attribute to individuals relying on the price of adjacent plots of
land as a guide.5,6 In their study of the disamenities caused by large-scale animal
operations and mushroom production, Ready and Abdalla (2005) account for spatial autocorrelation in their hedonic house price study.
The contribution of this paper is to extend the hedonic model of farmland values
to one in which spatio-temporal relationships are present. More specifically, the
model presented here assumes that the price of farmland is affected by a spatially
weighted average of sale prices achieved during past time periods, arguing that Patton and McErlean’s implicit assumption of a simultaneous spatial lag is an implausible representation of the market price formation process. Individuals
contemplating the purchase of farmland can use only the past as a guide, not the
future. The model presented in this paper further allows for the possibility that the
characteristics of spatio-temporally lagged sales might be important in determining
the extent to which they serve as comparables. In making these changes, the paper
pays much closer attention than did previous researchers to the manner in which
land markets appear to function.
Anticipating the main results of the paper, adding spatio-temporally lagged values
of farmland values as well as terms describing the characteristics of those sales
generates a highly significant improvement in explanatory power. At the same time,
that statistical significance of key farmland attributes appears to change. Finally,
calculating the value of farmland amenities is far easier than in a model characterised by a simultaneous spatial lag. Such findings are of great concern to those seeking to use hedonic analysis to measure the value of farmland attributes accurately.
It is important to note that although it has a temporal component, the purpose of
this paper is not to explain the evolution of farmland prices over time. That is the purpose of models methodologically quite different in outlook to the one presented here
in which dummy variables are used to absorb all macroeconomic influences. The focus
is purely on hedonic models and how previously unexplained cross-sectional variation
in farmland values can be reduced and the risk of potential bias lessened by including
spatio-temporally lagged values of the dependent and independent variables.7
The remainder of the paper is organised as follows. Section 2 offers a suitably
brief review of the relevant aspects of spatial econometrics while at the same time
directing the reader to more detailed references in the literature. Section 3 describes
the dataset upon which the model is estimated. Section 4 presents a series of
5
See also Maddison (2004) and Patton and McErlean (2004).
There are also a number of spatial analyses or discussions relating to the applicability of
spatial techniques to agriculture but not involving farmland values. Examples include Bockstael (1996), Weiss (1996), Nelson and Hellerstein (1997), Irwin and Bockstael (2002) and Roe
et al. (2002). Holloway et al. (2007) provide an excellent survey of the literature. Florax and
Van der Vlist (2003) give an overview of the more general application of spatial econometric
techniques.
7
Numerous papers have analysed the determinants of aggregate land prices using time series
data (e.g. Lloyd et al., 1991).
6
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David Maddison
econometric models and section 5 presents a new spatio-temporal model of farmland values, which is then formally compared with a model characterised by a
simultaneous spatial lag. The final section concludes.
2. Spatial Econometrics
Given the several recent reviews of the spatial econometrics literature, the review
provided here will be brief.8
Spatial data may present itself in a variety of forms. Data can be observed at
irregularly spaced points on a two-dimensional surface. Alternatively, spatial data
can refer to geographical areas, some of which may be contiguous to other units.
Spatial relationships in regression analyses involving such data can also be modelled in a variety of ways. Spatial lags hypothesise that the value of the dependent variable observed at a particular location is partially determined by a spatially weighted
average of the value of the dependent variable as measured at other locations. Such a
model cannot be estimated by OLS because of the problem of simultaneity bias and
must instead be dealt with using either instrumental variable estimators or maximum
likelihood techniques. The spatial lag model in matrix form is given by:
Y ¼ a þ Xb þ qWY þ e
ð1Þ
whereY is a (n · 1) vector of dependent variables, X a (n · k) matrix of explanatory
variables, W is the (n · n) spatial weight matrix, a and q are scalar parameters and
b is a (k · 1) vector of parameters.
A number of different assumptions can be made concerning the spatial weights
matrix required to compute the spatially weighted values at each location. One possibility is that the weights matrix should merely identify the nearest neighbour or n
nearest neighbours. An alternative is that the weights matrix should identify those
observations located within an arbitrarily fixed distance. A third alternative is that
the weights matrix contains the inverse geographical distances between observations
or some function thereof. In instances in which data refer to geographical areas as
opposed to points on a plane it is customary to construct a contiguity matrix indicating which geographical areas are adjacent to one another. Weights matrices are
commonly row-standardised such that their rows sum to unity and have zeros along
the leading diagonal.
Calculating the impact of marginal changes in the value of the independent variables in the spatial lag model is not as straightforward as it might first appear. The
reason is that changes in the independent variables have both direct and indirect
feedback effects on the value of the dependent variable. This fact can best be appreciated by solving the spatial lag model for Y to obtain:
Y ¼ ðI qWÞ1 Xb þ ðI qWÞ1 e
ð2Þ
This is an important point to which I shall return.
8
For a straightforward introduction to spatial econometrics, see LeSage (1997) and Dubin
(1998). Detailed texts include Cliff and Ord (1981), Upton and Fingleton (1985), Anselin
(1988) and Cressie (1993). A collection of interesting papers can be found in Anselin and Florax (1995). Anselin (2002) and Florax and Van der Vlist (2003) provide a recent overview of
the literature. An important recent contribution can be found in Anselin (2003).
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175
An alternative form of spatial relationship occurs when the dependent variable
can also be predicted as a function of spatially lagged values of the independent
variables. Unlike spatial lags, this model can be estimated using OLS. The spatial
regression model is given by:
Y ¼ a þ Xb þ WXc þ e
ð3Þ
where c is a (k · 1) vector of coefficients.
The final way of incorporating spatial relationships is through spatial dependence
in the error term. This can be in the form of spatially autoregressive errors or a spatial moving average. The spatial error model is given by:
Y ¼ a þ Xb þ e
ð4Þ
e ¼ kWe þ u
ð5Þ
where
and k is a scalar parameter. Such models can be estimated through feasible generalised least squares or maximum likelihood techniques.
A distinction can be drawn between spatio-temporal and simultaneous spatial
relationships. In spatio-temporal models, the observations are ordered in time such
that spatially lagged values of current dependent and independent variables are
allowed to influence only the future realisations of random variables. For the purposes of this paper, the distinction between these two types of model is very important. Calculating the marginal impact of changes in the independent variables is
also much easier in the context of a spatio-temporal model.9 (For an introduction
to spatio-temporal models, see Pace et al., 1998b.)
Spatial relationships in the data typically violate the assumptions underlying
OLS, leading either to inefficiency and invalid hypothesis testing procedures or even
to bias and inconsistency in the parameter estimates. The specification of spatial
models can proceed either on the basis of diagnostic tests or on the basis of theoretical reasoning. Another important consideration is that simultaneously modelling
several forms of spatial relationships such as the spatial lags and a spatially autoregressive error term is not possible at present. A final consideration is that whereas
several tests are available that indicate whether spatial relationships are present in
data, some of these tests have difficulty in discriminating between different forms of
spatial relationship.
3. Data
The dataset used to test for the presence of spatio-temporal relationships in agricultural land prices and to illustrate the consequences of failing to account for such
relationships comprises observations on farmland transactions that occurred in England and Wales from the beginning of January 1994 to the end of June 1996. These
are taken from the twice-yearly publication Farmland Market. In total, 601 observations are available for analysis although in order to generate a spatio-temporal lag
(see below) the first six months worth of data (94 observations) are kept in reserve.
9
This is of obvious concern to anyone wishing to calculate the implicit price of farmland
characteristics in hedonic price models. Maddison (2004) provides more details.
2008 The Author. Journal compilation 2008 The Agricultural Economics Society.
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David Maddison
A shortcoming of the dataset is that only those sales taking place through public
auction are included in the analysis. But as a record of farms sold through public
auction, the dataset is almost complete.10 Moreover, the sale of land through
public auction rather than private transaction generates precisely the kind of information externality that was earlier argued to give rise to spatio-temporal relations
in the data.
The dataset includes the current sale price of the farmland (PRICE) measured in
pounds sterling and also the size of the farm (ACRES) in acres. Many sales have
large farmhouses attached to them. Usually information is available only on the
number of bedrooms, and the presence of other agricultural buildings is not
recorded systematically. The number of bedrooms is divided by the number of acres
(BEDS ⁄ ACRES) to provide an indicator of the level of structural attributes per unit
of land. Animals and agricultural equipment are typically sold separately from the
land and its buildings. The observations are geographically referenced using the Ordnance Survey gazetteer (a database of all place names in Britain and their spatial
coordinates). Observations, whose location could not be determined to a satisfactory
degree of precision, were discarded. Repeat sales of land were also discarded.
Consistent with previous studies, a variable is included denoting the distance in
kilometres to the nearest city, partly as a measure of distance to market and partly
as a measure of development potential. A city is variously defined with at least
250,000 people to obtain the variable DIST250 or with at least 100,000 people to
obtain the variable DIST100. Note that there are 17 cities with a population size in
excess of 250,000 in Great Britain and 57 cities with a population size in excess of
100,000. A referee remarks that proximity to smaller urban areas might also be
important. The problem about including a variable indicating proximity to urban
areas with a population of significantly less than 100,000 is that the variation in the
resulting variable is sharply diminished. Observe already the difference in the standard deviation between variables DIST250 and DIST100 in Table 1. Alternately
put, nowhere in England or Wales is very far from a community of 10,000 or more.
Following Shi et al. (1997), I also create a variable URBAN which takes into
Table 1
The dataset
Variable
PRICE ⁄ ACRES
ACRES
BEDS
ALC
DIST250
DIST100
URBAN
Mean
SD
Minimum
Maximum
2,814
84
1.14
3.00
57
36
5,867
1,441
148
2.06
0.82
38
20
2
154
7
0
1
5
2
5,866
12,571
2,794
11
5
211
123
5,900
10
Even if this dataset were extended to include farms sold by private treaty, it would, of
course, still represent only a small fraction of the farmland in England and Wales. This is a
problem of most hedonic analyses, although see Koundouri and Pahshardes (2003) for an
exception.
2008 The Author. Journal compilation 2008 The Agricultural Economics Society.
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177
account both geographical distances and the population size of nearby cities. This
involves calculating for each plot i and for each city j of at least 100,000 people the
following variable:
X
URBANi ¼
POPj =DIST2ij
ð6Þ
j
where POP is the population of city j · 100,000 and DIST is the distance between
plot i and city j in kilometres. Including these variables does not of course preclude
the possibility that the spatial techniques employed below will prove better at modelling the twin influences of distance to market and development potential.
A series of land-quality variables were matched to the data using the geographical
coordinates of the farmland sold. The 5-km grid square agricultural land classification (ALC) system of England and Wales classifies land into one of five different
grades (Ministry of Agriculture Food and Fisheries, 1988). The land-grading system
is based on the extent to which among other things, climate, physical composition,
drainage characteristics and slope impose long-term limitations on lands’ agricultural use. Grade 1 agricultural lands are the best and grade 5 the worst. The same
dataset also records if land was non-agricultural in nature or simply not surveyed in
which case it is dropped from the dataset.
The obvious disadvantage of the ALC is that it fuses together a number of characteristics, making it impossible to determine the individual contribution any given
characteristic makes to farmland productivity. The advantage of using the classification is that it is the result of many years of fieldwork and is available at a high level
of spatial resolution. As using data at a low level of spatial resolution can itself
result in spatial autocorrelation and because the focus of this paper is primarily
methodological, on balance it was deemed appropriate to use the classification as
the sole measure of farmland productivity.
The most natural way of incorporating the land quality data is obviously to
include four dummy variables corresponding to the different land classifications,
along with the constant term. But in what follows, a single variable taking integer
values from 1 to 5 was included in the model. This is because the majority of land
is classified as either grade 3 or 4 and because including separate dummies for each
land classification did not result in a statistically significant improvement in fit in
any of the regression models presented below.
Finally, dummy variables DUM1, DUM2 and DUM3 identify the periods January 1995 to June 1995, July 1995 to December 1995 and January 1996 to June
1996, respectively. The constant term refers to the period July 1994 to December
1994. The purpose of these dummy variables is to account for changes in the current price of inputs and outputs as well as future expectations regarding these
things. Changes in price expectations are clearly very important in determining the
value of land and failing to control for these risks obscuring the value of farmland
attributes. The dataset is described in Table 1.
The Euclidean distance between each parcel of land is computed and used to construct a distance matrix D. The median distance between farmland sold by auction
during the period July 1994 to June 1996 is 182 km. The minimum distance is 1 km
and the maximum distance is 644 km. The largest minimum distance is 74 km (i.e.
one farm is 74 km from its nearest neighbour) and the smallest maximum distance
is 329 km (i.e. one centrally located farm is within 329 km of every farm in the
dataset). In order to construct the weights matrix, W, required for the ensuing
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David Maddison
calculations, it is convenient to assume that the individual elements (wij) of the
weight matrix evolve according to:
f
wij ¼ 1=d ij
ð7Þ
where the exponent f is often referred to as the friction parameter and dij are the
elements of D.11 The leading diagonal of the weights matrix W is filled with zeros
and the matrix is then row-standardised such that the rows of the matrix each sum
to unity. An alternative assumption is to compute a standardised binary spatial
weight matrix identifying those farms which lie within a pre-specified distance of
one another. Unfortunately, the largest minimum distance between sales contained
in the dataset (74 km) limits the usefulness of this approach. But by increasing the
size of the friction parameter, the weight placed on more distant farms can be made
arbitrarily small and does away with the need to specify a distance limit. I have
examined different values for f and find that the inverse distance model with f = 1
yields the most pronounced evidence of spatial dependence.
4. Econometric Results
The presence of spatial autocorrelation in the data is confirmed using Geary’s
C-statistic (Geary, 1954). The results of these tests are presented in Table 2 and
indicate that as might be anticipated, the variables PRICE ⁄ ACRES and ALC are
significantly autocorrelated over space. The level of structural attributes per unit of
land BEDS ⁄ ACRES is spatially autocorrelated at the 5% level of confidence and
ACRES is not spatially autocorrelated. The fact that some explanatory variables
are highly spatially autocorrelated, whereas others are not, suggests that accounting
for spatial effects might have a differential impact on the perceived size and statistical significance of these variables.
Using an alternative approach to examine the range of spatial autocorrelation, a
series of binary standardised spatial weight matrices were created identifying neighbouring farms that fall within, e.g. 0–75, 75–150, 150–225 km of one other. Computing Geary’s C-statistic for each of these binary spatial weight matrices suggests
that the price per acre of farms within 150 km of each other exhibits statistically
significant positive spatial autocorrelation and that for farms within 75 km of each
other this statistical relationship is highly significant. This same test is also conducted using cumulated distance, i.e. for farms that fall within 0–75, 0–150,
Table 2
Global spatial autocorrelation statistics
Variable
PRICE ⁄ ACRES
ALC
BEDS
ACRES
Geary’s C
Probability
0.939
0.846
0.978
1.042
0.000
0.000
0.010
0.253
Note: Spatial weight matrix = inverse distance row-standardised.
11
See, for example, the STATA Technical Bulletin STB-60 published in March 2001.
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A Spatio-temporal Model of Farmland Values
Table 3
Banded measures of spatial correlation
Distance (km)
0–75
75–150
150–225
225–300
300–375
375–450
450–525
525–600
Geary’s C
)5.493
)1.828
)0.275
1.298
)0.078
)0.650
)0.657
)0.534
(Prob
(Prob
(Prob
(Prob
(Prob
(Prob
(Prob
(Prob
=
=
=
=
=
=
=
=
Distance (km)
0.000)
0.034)
0.392)
0.097)
0.469)
0.258)
0.256)
0.297)
0–75
0–150
0–225
0–300
0–375
0–450
0–525
0–600
Geary’s C
)5.493
)3.940
)1.815
0.020
1.276
1.095
0.903
0.729
(Prob
(Prob
(Prob
(Prob
(Prob
(Prob
(Prob
(Prob
=
=
=
=
=
=
=
=
0.000)
0.000)
0.035)
0.492)
0.101)
137)
0.183)
0.233)
Notes: Spatial weight matrix = inverse distance row-standardised.
Variable = PRICE ⁄ ACRE.
0–225 km of one another. This suggests spatial autocorrelation even up to a range of
225 km. Note that the probability values given in Table 3 refer to a one-tailed test.
To begin with, various special cases of the following model are estimated:
log(PRICE/ACRES) ¼ a þ b1 ACRES þ b2 ALC þ b3 (BEDS/ACRES)
þ b4 (BEDS/ACRES)2 þ b5 DIST250 þ b6 DIST100
þ b7 URBAN þ b8 DUM1 þ b9 DUM2
X
þ b10 DUM3 þ q
wi log(PRICE/ACRES)i þ e
where:
e¼k
X
w i ei þ u
ð8Þ
ð9Þ
and a, b1–10, q and k are the parameters. Note that the dependent variable was
taken as the log of PRICE ⁄ ACRES following attempts to estimate a Box–Cox
transformation of the dependent variable. Taking PRICE ⁄ ACRES as a purely linear variable resulted in an equation which did not pass the RESET test for functional form (Ramsey, 1969). For the same reason, it was necessary to add a
quadratic term for BEDS ⁄ ACRES. The number of acres was included as an explanatory variable in case repackaging of land is a costly activity.12
Setting the parameters q and k equal to zero yields the usual hedonic price equation estimable by OLS. Allowing the parameter q to vary results in a model in
which individual sale prices are affected by the prices of neighbouring properties.
Allowing the parameter k to vary results in a model where the residuals are autocorrelated over space.13
The regression results are presented in Table 4 in which the conventional hedonic
model estimated by OLS with q and k are set equal to zero. Note that the t-statistics
12
The so-called ‘linearity hypothesis’ (Parsons, 1990) would imply taking the dependent variable PRICE in levels and multiplying any environmental attributes like ALC by the number
of acres. Here I prefer to take the dependent variable as price per acre in order to reduce heteroscedasticity so I divide the structural attribute BEDS by the number of acres.
13
All calculations are undertaken using STATA.
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Table 4
The regression results for the OLS model
Variable
CONSTANT
ACRES
ALC
BEDS ⁄ ACRES
(BEDS ⁄ ACRES)2
DIST250
DIST100
URBAN
DUM1
DUM2
DUM3
F-statistic
No. obs.
R2
RESET
Parameter (t-stat)
61.819
)0.000793
)0.132
11.359
)20.609
)0.00281
)0.00142
)0.00913
0.2183
0.295
0.355
87.74
507
0.407
0.05
(1.81)
()6.31)
()5.98)
(13.84)
()10.24)
()3.80)
()1.28)
()1.57)
(4.64)
(5.83)
(6.99)
(Prob=0.000)
Parameter (t-stat)
8.165
)0.000794
)0.137
11.288
)20.591
)0.00303
(Prob=0.983)
0.220
0.297
0.352
103.38
507
0.405
0.11
(110.69)
()6.34)
()6.34)
(13.80)
()10.28)
()4.89)
(4.68)
(5.87)
(6.97)
(Prob=0.000)
(Prob=0.951)
Notes: Dependent variable = log(PRICE ⁄ ACRES).
Method = OLS.
t-statistics are heteroscedastic-consistent.
are based upon robust standard errors. The results reveal that as land quality worsens the price of land declines. The results also indicate that the greater the number
of bedrooms per acre the greater the price albeit at a diminishing rate. Smaller plot
sizes command a higher price per acre indicating that the repackaging of land is a
costly activity. Turning now to the alternative measures of distance to market cum
urban development potential, defining a city in terms of an urban area with a
population in excess of 250,000 is by far the most successful strategy in terms of fit.
This variable is negatively signed as anticipated, and highly significant. By contrast,
DIST100 and URBAN are jointly insignificant and dropped from all further
regressions.14
Before conducting any spatial regressions, an important issue to address is
whether the regressions are stable across different regions or whether geographically
distinct markets exist, within which the implicit prices of farmland characteristics
differ (Straszheim, 1974). Freeman (1993) explains the circumstance under which
geographically segmented markets arise: when buyers do not, for whatever reason,
participate in different markets while at the same time these regions differ in terms
of the supplies of and demands for attributes.
When markets are segmented but model intercepts and coefficients are constrained to be the same, market segmentation can manifest itself in positive tests for
spatial autocorrelation. There is however no prior information to indicate the areas
that might potentially be considered separate markets, and furthermore the sample
size is of modest dimensions. These considerations suggest that the most appropriate strategy is to use the dummy variable approach to allow the intercept and slopes
of the regression equations to differ across each of four quadrants referring to the
14
F(2, 496) = 1.57.
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Table 5
The regression results for the OLS model with regional land markets
Variable
SE parameter
(t-stat)
NE parameter
(t-stat)
NW parameter
(t-stat)
CONSTANT
7.708 (43.81)
8.0380 (55.88)
8.690
ACRES
0.000117 (0.31) )0.000901 ()5.55) )0.000678
ALC
)0.0281 ()0.58)
)0.116 ()2.86)
)0.259
BEDS ⁄ ACRES
11.285 (3.56)
11.621 (9.13)
10.960
(BEDS ⁄ ACRES)2 )6.212 ()0.22)
)22.526 ()7.47)
)23.920
DIST250
0.000127 (0.09)
0.000004 (0.00)
)0.00343
DUM1
0.0901 (1.14)
0.269 (3.17)
0.247
DUM2
0.167 (2.21)
0.149 (1.55)
0.278
DUM3
0.147 (1.71)
0.380 (4.16)
0.428
F-statistic
50.52 (Prob=0.000)
No. obs.
507
R2
0.486
RESET
1.34 (Prob=0.260)
SW parameter
(t-stat)
(39.20)
7.911 (55.12)
()2.27) )0.000944 ()1.47)
()4.03)
)0.0902 ()2.41)
(2.89)
7.172 (2.00)
()0.55)
24.868 (0.78)
()3.12) )0.00321 ()2.93)
(2.08)
0.1403 (1.66)
(2.06)
0.503 (5.41)
(3.59)
0.424 (4.51)
Notes: Dependent variable = log(PRICE ⁄ ACRES).
Method = OLS.
t-statistics are heteroscedastic-consistent.
south-east (SE), the north-east (NE), the north-west (NW) and the south-west
(SW). The coefficients relating to these geographically defined sub-markets are presented in Table 5. The test for parameter stability is significant at the 1% level of
confidence, implying that the hypothesis of a market for farmland, unified by trade
in agricultural commodities, must be abandoned.15 Across the four sub-markets,
several important differences are observed in the implicit values of farmland characteristics. In the SE and the SW, the price per acre is not affected by farm size. In
the SE, high-quality farmland does not attract a premium possibly because it is relatively more abundant there. Alternatively, the value of agricultural land grade 3, 4
and 5 may be higher in the SE because of the greater developmental opportunities
typically denied to prime agricultural land or because of the higher recreational and
environmental values in part related to the higher population and greater wealth of
the SE. Finally, distance to the nearest big city is an amenity only for farms in the
NW and SW of the country.
Table 6 displays test results for spatial error and spatial lag dependence in the
OLS model with heterogeneous land markets. The Lagrange multiplier tests are less
useful in this respect because each test has some power against the alternative form
of spatial dependence. The robust Lagrange multiplier tests overcome this problem
(Anselin et al., 1996). Both the robust and the non-robust versions of the test statistics point to the existence of a spatial lag.16 There is no evidence that spatial autocorrelation is due to omitted neighbourhood characteristics (the tests for spatial
error dependence are statistically insignificant).
15
F(27, 471) = 3.92.
Note that there are other possible causes of a significant spatial correlation, e.g. use of an
incorrect functional form.
16
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David Maddison
Table 6
Tests of spatial autocorrelation in the OLS model
Spatial error dependence
Moran’s I-statistic
Lagrange multiplier test
Robust Lagrange multiplier test
Spatial lag dependence
Lagrange multiplier test
Robust Lagrange multiplier test
3.631 (Prob=0.000)
2.191 (Prob=0.139)
2.910 (Prob=0.088)
9.134 (Prob=0.003)
9.852 (Prob=0.002)
Note: Spatial weight matrix = inverse distance row-standardised.
Table 7
The regression results for the spatial lag model
Variable
SE parameter
(T-stat)
NE parameter
(T-stat)
NW parameter
(T-stat)
CONSTANT
2.013 (1.21)
2.320 (1.39)
2.866
ACRES
0.000102 (0.28) )0.000910 ()5.65) )0.000635
ALC
)0.0237 ()0.50)
)0.108 ()2.81)
)0.252
BEDS ⁄ ACRES
11.172 (3.53)
11.253 (9.23)
10.833
(BEDS ⁄ ACRES)2
)2.724 ()0.10) )21.0168 ()7.19) )21.969
DIST250
0.000440 (0.31)
0.000313 (0.31)
)0.00206
DUM1
0.0983 (1.28)
0.252 (3.13)
0.243
DUM2
0.180 (2.43)
0.144 (1.56)
0.286
DUM3
0.164 (1.93)
0.367 (4.31)
0.410
q
0.721 (3.43)
Log-likelihood
)198.0011
No. obs.
507
Square correlation
0.498
(pseudo-R2)
SW parameter
(T-stat)
(1.68)
2.239 (1.35)
()2.31) )0.00102 ()1.72)
()4.17) )0.0772 ()2.13)
(3.03)
6.113 (1.83)
()0.53)
34.252 (1.18)
()1.89) )0.00263 ()2.45)
(2.18)
0.150 (1.90)
(2.27)
0.286 (2.27)
(3.64)
0.425 (4.72)
Notes: Dependent variable = log(PRICE ⁄ ACRES).
Method = ML (Maximum Likelihood).
Spatial weight matrix = inverse distance row-standardised.
t-statistics are heteroscedastic-consistent.
Table 7 presents the same regression equation but with spatial lag dependence
(q allowed to vary but k is set equal to zero). The parameter estimates are not much
altered but the spatial autocorrelation parameter q is highly significant.17 The
results from this model are very similar to those from the heterogeneous land market model without a spatial lag except that distance to the nearest big city is no
longer significant in the NW whereas previously it was significant at the 1% level of
confidence. But it is important to avoid believing that because the parameter coefficients are similar that the implicit value of marginal changes in the value of the
dependent variables must also be similar. The preceding section illustrated that
17
Note that I do not allow this parameter to vary over sub-markets in the specification
shown in Table 7.
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183
calculating the marginal value of changes in the dependent variable in a model characterised by a spatial lag is quite involved.
5. A Spatio-temporal Model
The chief limitation of the spatial lag model presented in Table 7 which has so far
characterised the literature is that it ignores entirely the temporal nature of the
data at the expense of focusing on the spatial aspects. But as hedonic datasets are
almost invariably composed of a sequence of sales, the assumption of a simultaneous spatial lag present in earlier analyses is an impossible representation of the
price formation process. In effect, it suggests that the price of a plot of farmland
is determined by the price of plots as yet unsold. More plausible is to assume that
farmland prices depend on the price of recently sold farmland and, in particular,
on the price of recently sold farmland in the immediate vicinity.18 This suggests
including the spatio-temporally weighted values of the dependent variable as an
explanatory variable and this in turn requires the creation of a spatio-temporal
weight matrix.
The role of the spatio-temporal weight matrix is to identify inverse distancebased standardised weights for each observation based upon a temporally defined
subset of the observations. This matrix Z is obtained from the Hadamard product
of matrices W (previously defined) and T. If sales are ordered by time, then matrix
T is lower triangular and comprises a null matrix interspersed with blocks of ones
identifying for each observation on farmland sold in an earlier time period. The
construction of this matrix also involves making a decision regarding the length of
the time period over which individuals base their information. Because the dataset
here is of modest dimensions, it is arbitrarily assumed that farmland values are
determined by spatially weighted sale prices realised in the preceding six months.19
Finally, the matrix Z is once more row-standardised. Most of the computational
effort involved in spatio-temporal analysis goes into the construction of this
matrix.
A second limitation of the spatial lag model is that it assumes that the price
of farmland is affected by the price per acre of nearby land but not by the characteristics of the land that was sold. In such a model, individuals risk basing
their judgements on land sales that are not comparable in terms of characteristics.20 This deficiency is easily remedied by including the spatio-temporally lagged
values of farmland characteristics as additional explanatory variables. Including
18
There is one way of rescuing the simultaneous spatial lag model. That is, if the same market participants are involved in the bidding. The fact that we use data observed at six-month
intervals would seem to diminish this possibility. The simultaneous spatial lag model might
also be appropriate if farmers were asked about their perceptions of the worth of their land
at a particular point in time.
19
Without a far longer time period, it is not possible to answer the question: How far back
in time do buyers make comparisons? Presumably, sales in the recent past carry most weight.
20
The information conveyed by the fact that someone was prepared to pay £10,000 per acre
for a plot of nearby land is for example quite different depending on whether this land was
grade 1 agricultural land with a five-bedroom farmhouse or a plot with grade 4 agricultural
lands and no dwelling.
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David Maddison
the spatio-temporally lagged values of the dependent and independent variables
as advocated by Pace et al. (1998b) and Can and Megbolugbe (1997) yields the
following equation:
log(PRICE/ACRES) ¼ a þ b1 ACRES þ b2 ALC þ b3 (BEDS/ACRES)
þ b4 (BEDS/ACRES)2 þ b5 DIST250 þ b6 DUM1
X
zi log(PRICE/ACRES)i
þ b7 DUM2 þ b8 DUM3 þ q
X
X
þ c1
zi ACRESi þ c2
zi ALCi
X
X
þ c3
zi (BEDS/ACRES)i þ c4
zi (BEDS/ACRES)2i
X
þ c5
zi DIST250i þ e:
ð10Þ
This model is estimated by OLS jointly for each sub-market and, from the results
in Table 8, it emerges that the spatio-temporally lagged variables are significant as a
group at the 1% level of confidence.21,22 Furthermore, there is no evidence of any
outstanding spatial autocorrelation in the residuals in the regression. Moreover,
compared with the model in Table 7 the squared correlation (or R2) statistic has
risen from 0.49 to 0.52.
Apart from the greater inherent plausibility of the spatio-temporal model and the
evident importance of accounting for spatio-temporally lagged values of the dependent variables there are two other advantages of the spatio-temporal model. First,
unlike the spatial lag model, it is easy to calculate the implicit value of marginal
changes in the level of the characteristics as there is no simultaneity present.23 Second, the model can be estimated using OLS.
The main reason for being concerned about spatial autocorrelation in hedonic
datasets is to restore the optimal properties of the estimation techniques. This has
been achieved by including a set of spatio-temporal variables whose contribution to
the explanatory power of the regression equation was found to be highly significant
(Lagrange multiplier tests reveal no trace of spatial autocorrelation). It remains only
to compare the parameter estimates from the spatio-temporal model with those
taken from the heterogeneous market model presented in Table 5 in order to determine what if anything is changed.
The coefficient on ACRES in the NW quadrant is now insignificant at the 5%
level of confidence. The coefficients on ALC remain statistically significant at the
5% level of confidence for the NE, NW and SW. The coefficient on BEDS ⁄ ACRES
becomes statistically insignificant for the SW. Of particular interest is the fact that
distance to the nearest large city, which was highly significant in both the NW and
SW is now no longer significant. The correct interpretation is presumably not that
21
Because of the absence of simultaneity, such models can be consistently estimated by OLS.
F(24, 447) = 2.95.
23
For example, Table 8 immediately indicates that there is a 10.5% difference in land values
between grade 2 and grade 3 agricultural lands in the NE, and a 24.9% difference in the
NW. The inclusion of spatio-temporal terms does not complicate the calculation of implicit
prices and serves only to eliminate any potential bias that might otherwise arise. For a discussion relating to the calculation of the implicit value of amenities in the spatial lag model,
see Maddison (2004).
22
2008 The Author. Journal compilation 2008 The Agricultural Economics Society.
The regression results for the spatio-temporal model
Variable
CONSTANT
ACRES
ALC
BEDS ⁄ ACRES
(BEDS ⁄ ACRES)2
DIST250
DUM1
DUM2
DUM3
P
Pzi · log(PRICE ⁄ ACRE)i
Pzi · ACRESi
Pzi · ALCi
Pzi · BEDS ⁄ ACRESi 2
Pzi · (BEDS ⁄ ACRES) i
zi · DIST250i
F-statistic
No. obs.
R2
RESET
SE parameter (T-stat)
)8.546
)0.000162
)0.0340
11.715
)6.306
0.00181
0.689
)0.377
)0.516
2.120
)0.000271
0.140
66.556
)1274.378
)0.00541
()1.31)
()0.43)
()0.79)
(4.16)
()0.23)
(0.91)
(2.58)
()2.24)
()2.05)
(2.40)
()0.07)
(0.67)
(1.51)
()2.23)
()0.54)
NE parameter (T-stat)
12.631
)0.000920
)0.105
11.167
)20.857
0.00241
0.421
0.396
0.473
)0.303
)0.00562
)0.354
19.784
)153.970
)0.0221
NW parameter (T-stat)
(2.09)
7.974
()6.15)
)0.000452
()2.19)
)0.249
(7.54)
9.999
()5.62)
)2.482
(1.74)
0.00118
(3.56)
0.2411
(1.71)
0.228
(2.32)
0.150
()0.41)
0.354
()1.62)
)0.00150
()1.41)
)0.562
(1.00)
28.0051
()1.12)
)36.00454
()2.44)
)0.0169
36.88 (Prob=0.000)
507
0.523
1.68 (Prob=0.171)
(1.22)
()1.67)
()3.39)
(2.66)
()0.05)
(0.46)
(1.55)
(1.01)
(0.64)
(0.51)
()1.48)
()0.83)
(2.32)
()0.46)
()1.75)
SW parameter (T-stat)
7.489
)0.000982
)0.0803
5.474
45.0216
)0.00338
0.0787
0.482
0.383
0.202
0.00323
)0.428
)24.343
250.238
0.000870
(2.32)
()1.46)
()1.98)
(1.49)
(1.38)
()1.49)
(0.68)
(3.51)
(1.54)
(0.50)
(0.97)
()1.88)
()0.92)
(0.84)
(0.17)
A Spatio-temporal Model of Farmland Values
2008 The Author. Journal compilation 2008 The Agricultural Economics Society.
Table 8
Notes: Dependent variable = log(PRICE ⁄ ACRES).
Spatio-temporal weight matrix = inverse distance row-standardised.
t-statistics are heteroscedastic-consistent.
185
186
David Maddison
distance to market or development potential are unimportant, but rather that
including spatio-temporally lagged variables does a better job of absorbing the variation in the data caused by proximity to the market and development potential than
distance to the nearest big city. Such findings draw attention to the importance of
the spatio-temporal modelling techniques outlined in this paper.
Despite the fact that they are highly significant as a group, the coefficients on the
spatio-temporal variables have relatively large standard errors. The reason is that
the spatio-temporal lags in the dependent and independent variables are somewhat
correlated with each other. The coefficients on the spatio-temporal terms are not,
however, the focus of attention in a hedonic regression. I attempted to simplify the
model by excluding the spatio-temporally lagged values of the independent variables, but this restriction is rejected at the 1% level.24 This highlights the importance of controlling for comparability.
Spatio-temporal models can be extended in a number of directions. It is in principle possible to increase the number of spatio-temporal lags to include sales from
earlier time periods which is not done here because of the relatively short time period covered by the dataset. In the same way that buyers apparently take relatively
greater note of nearby properties, it seems probable that they also take relatively
greater note of more recent sales.
6. Conclusions
This paper has presented a spatio-temporal model in which farmland values are
affected not only by the price per acre of nearby land sold in the preceding six
months but also by the characteristics of the land that was sold. This paper builds
on earlier research indicating that agricultural land prices might be characterised by
a spatial lag. Compared with the spatial lag model, the formulation in this paper
appears to possess a number of advantages. First it acknowledges the role of comparables in determining prices. Second, it can be estimated using OLS and third it
makes it easier to calculate the implicit price of farmland characteristics as there is
no simultaneity present.
Accounting for spatio-temporal relationships also appears to have some impact
on the perceived value and statistical significance of the characteristics of farmland.
This is not surprising given the fact that some of the characteristics of farmland are
very highly spatially correlated. The spatio-temporal model also appears to do a
better job of controlling for hard to model but conceptually important local influences such as distance to market and development potential. The spatio-temporal
nature of farmland sale price data should not be ignored in empirical studies. Likewise, it is probable that spatio-temporal relationships are also important in other
hedonic applications.
Acknowledgements
The author would like to acknowledge the helpful comments of the editor and
two anonymous referees. Any errors are however the sole responsibility of the
author.
24
F(20, 447) = 2.15.
2008 The Author. Journal compilation 2008 The Agricultural Economics Society.
A Spatio-temporal Model of Farmland Values
187
References
Anselin, L. ‘Model validation in spatial econometrics: A review and evaluation of alternative
procedures’, International Regional Science Review, Vol. 11, (1988) pp. 279–316.
Anselin, L. ‘Under the hood issues in the specification and interpretation of spatial regression
models’, Agricultural Economics, Vol. 27, (2002) pp. 247–267.
Anselin, L. ‘Spatial externalities spatial multipliers and spatial econometrics’, International
Regional Science Review, Vol. 26, (2003) pp. 153–166.
Anselin, L. and Florax, R. New Directions in Spatial Econometrics (Berlin: Springer Verlag,
1995).
Anselin, L., Bera, K., Florax, R. and Yoon, M. ‘Simple diagnostic tests for spatial dependence’, Regional Science and Urban Economics, Vol. 26, (1996) pp. 77–104.
Basu, S. and Thibodeau, T. ‘Analysis of spatial autocorrelation in house prices’, Journal of
Real Estate Finance and Economics, Vol. 17, (1998) pp. 61–86.
Bell, K. and Dalton, T. ‘Spatial economic analysis in data-rich environments’, Journal of
Agricultural Economics, Vol. 58, (2007) pp. 487–501.
Benirschka, M. and Binkley, J. ‘Land price volatility in a geographically dispersed market’,
American Journal of Agricultural Economics, Vol. 76, (1994) pp. 185–195.
Bockstael, N. ‘Mo