A multiple indicators and multiple causes model for valuation

Value assessment (valuation) of certain real properties is commonly required for a variety of purposes. In this article an approach to define an estimator of the value of a real property, with good statistical properties is proposed. This estimator reaches a consensus between the various available methods. The estimator proposed uses all available information optimally, including the information which value indicators contain on the effect of contextual factors, and this furnishes it with a greater relative efficiency with regard to alternative estimators. It is also shown how the proposed approach is more general and flexible than the usual hedonic approach. The approach is illustrated using data on agricultural real properties and the relative efficiency of the compromise estimator is assessed using this same data. Additional key words: hedonic price modeling, land valuation, latent variables, measurement errors, unbiased and optimal linear estimators.


Introduction
Valuation (estimation of the value) of certain real properties has been necessary for a very long time, as an appraisal or basis of appraisal set by authorities for any purposes which may require this valuation (England, 2003).In the case of commodities, it is easy to find an objective standard on which to found the valuation process: the market price.However, in order to value real estates, such as an agricultural property, there are many possibilities (e.g.net realizable value, opportunity cost, or replacement cost), and there is disagreement over the best method to measure value (Lemke, 1966).
There is agreement in value theory on considering effective demand, scarcity of supply, exchange, and transferability as the sources and bases of value.However, there is disagreement about the definition of value, due to the fact that the attributes of value identified and emphasized by each school of economic thought have often focused on the central current social, political, and economics issues.This disagreement is a serious obstacle for the valuation process, since the def inition of value has a key role both in directing data collection, as well as estimation procedures leading to an appropriate estimation of value (Grissom, 1985).
To estimate the value of a given real property, the International Valuation Standards Committee (IVSC) suggests three different approaches: the sales comparison approach, the income capitalization approach and the cost approach (IVS, 2005).All three approaches presume that economic agents are rational and compare the benefits from buying or selling a property with investment alternatives.The economic rationale of the sales comparison approach is that no informed agent would pay more for a property than other agents have recently paid for comparable properties given that the general market conditions are the same.The economic rationale of the income approach for existing properties is that no economic agent will pay more for a property than he will retrieve by holding the property.Finally, the economic rationale of the cost approach is that no rational agent will pay more for an existing property than the cost of constructing an equally desirable substitute property (Corgel et al., 2001).
All three valuation approaches are consistent with economic reasoning and neither of these approaches is better than the other (Shiller and Weiss, 1999).The problem is that the appraiser has to make a f inal reconciliation step to reach a f inal estimate of the market value and no methodology exists on how to mix the different market values calculated by the three approaches into one final market value.
This paper focuses on a method to achieve a consensus among the various valuation methods proposed in valuation literature, in order to reach a final estimate of the market value of a given real property.This consensus should make optimal use of the available data.Instead of a value definition, a «valuation hypothesis» is proposed as a guide during the data collection process and the value estimation process.It is assumed that the value of a given property is more than a physical phenomenon -a quality inherent to the propertyand is a constructed human perception (Hadley, 1928).Human perceptions are to a large extent socially acquired, and to estimate the value of the given property, it is assumed that it is possible to establish a social consensus about the set of its measurable attributes of value, which here are called multiple-causes of value.
This «valuation hypothesis» is also assumed in the hedonic approach.Hedonic price modeling is a prominent tool in analyzing real estate prices, and it has been used for calculating real estate price indices (Bailey et al., 1963;Sheppard, 1999;Malpezzi, 2002) as well as calculating the impact on property values from proximity to a landfill and other sources of environmental risks (Freeman, 1979;Reichert et al., 1991;Nelson et al., 1992).The approach is also known as hedonic regression in the literature because it relates observed prices to property-specific characteristics by regression methods.It is the core assumption of the hedonic approach that prices are given by a function of property-specif ic attributes (Cropper et al., 1988), which will be called the hedonic assumption from now on.
In this paper, all approaches on the valuation problem suggested in the literature are taken into account, including the hedonic approach.Our interest focuses on how to best combine the value estimates given by the approaches proposed in the literature into one final market value estimate.In addition, our approach accounts for the effect due to contextual factor.Context can play a tremendously important role in human perception (Luke, 2004).This effect of unobserved contextual factors must be taken into account, in addition to property-specific characteristics, when the value of a real estate property is estimated.
In this study, a statistical approach is proposed to estimate the value of an agricultural property as the sum of two components: one stemming from measurable property-specific value attributes, and the other as a consequence of unobserved contextual factors, using data describing the indicators of value.The term «indicator» is used here to denote an uncertain measure of value, when this uncertainty is not measurable.In this sense, both subjective measures as well as some objective measures are considered indicators; for instance, the present discounted value of the future cash flow of the agricultural property being valued (capitalization approach) is considered an indicator, since there is uncertainty about the discount rate, and this uncertainty is rarely measurable.When it is possible to quantify the uncertainty of a measure of value, this measure will be called «estimate».
To derive both an estimator of the value of a given agricultural property and the standard error of this estimator, a multiple indicators and multiple causes model (MIMCM) is proposed (Robins and West, 1977;Skrondal and Rabe-Hesketh, 2004).It is a statistical model which describes the relationship between a set of indicators on the value of a given agricultural property (multipleindicators), and the measurable attributes (multiplecauses) of this property.This model allows us to «borrow strength» from data relative to properties related to the property to be valued, in order to obtain the best estimate of the value of the property under consideration.Based on this model, a compromise among the various valuation methods proposed in the valuation literature is established and this consensus should be that which makes an optimal use of the available data, relative to multiple indicators and causes (attributes).
This compromise valuation could be useful to reduce the traditional disagreement on the best method to measure a defined value.The procedure is illustrated using a case study from agriculture, but it can be used in the remaining economic sectors as differences among valuation methods used in each sector are of a more practical than theoretical nature (ASFMRA-AI, 2000;IVS, 2005).
The purpose of this study is to f ind a statistical approach to the problem of estimating the value of an agricultural property, and the approach suggested consists in expressing this value as the sum of two components.The first component stems from measurable property-specific value attributes, which is the usual procedure in the hedonic approach.The other component evaluates unobserved contextual factors by using data on the indicators of value, for example, valuations arising from methods suggested by the IVSC.

The model
The relationship between the indicator y ij of the value of an agricultural property i, and the measurable attributes of this property x i , is established using the structural model: where [1] is the behavioral equation and [2] is the measurement equation.
The behavioral equation specifies that the value y i is perceived as the sum of two components: one, x T i β, due to the measurable attributes included in the vector x i (β is the vector of parameters representing the changes on y i due to changes on x i ), and the other, v i , is the effect due to unobserved contextual factors.An agricultural property with x i attributes is expected to have a value Nevertheless, unobserved contextual factors can have an effect v i on value.For measurement purposes, it is considered that this effect is a realization of a random zero mean variable Ev i = 0, with variance σ 2 v and covariance null Cov(v i , v i' ) = 0 for i ≠ i'.In other words, it is assumed that because of unobserved contextual factors the value y i of a given agricultural property (i.e.given x i ) varies around its expected value with variance V(y The measurement equation specif ies that each indicator of y i in the set {y ij ; j = 1,2,...,J} deviates from y i in a certain (unobservable) quantity, e ij = y ij -y i , due to the difficulties involved in perceiving β and v i , as well as to measurement errors of the attribute x i .The disagreement between valuation measurement results is modeled through this deviation, assuming that this deviation is a realization of a random variable of mean zero, E(e ij |y i ) = 0 (so that E(y ij |y i ) = y i , and assuming the indicators are unbiased), and variance In addition, it is assumed that the deviation e ij is independent of the value y i , that is Cov(v i , e ij ) = 0, and that the deviation of two indicators are independent, Cov(e ij , e ij' ) = 0; ∀j ≠ j'.
Replacing [2] in [1] the reduced form is obtained, where where σ 2 v and σ 2 e are the so-called variance components.The contextual factors effect v i induces positive covariance among the indicators in the set {y ij ; j = 1,2,...,n i }, and the covariance structure is: The correlation coefficient between indicators of the value of a same property, , is a measurement of the «level of agreement» among indicators [it is called «reliability ratio» in the measurement error models theory (Fuller, 1987;Robinson, 1991) represent the proportion of the total variability V(y ij |x i ), due to the disagreement among indicators.In fact, in our approach any disagreement among the indicators {y ij ; j = 1,2,...,n i } of a given agricultural property i (i.e.given x i ) in a given context (i.e.given v i ) is considered to be due to the difficulties involved in perceiving β and v i and to measurement errors of the attribute x i , which is measured by σ 2 e .When there is total agreement among the indicators, then σ 2 e = 0 and ρ takes its maximum value, ρ = 1 (for any σ 2 v > 0).When the disagreement among indicators increases in such a way that the ratio approaches zero, then ρ decreases approaching its minimum value, zero.The «degree of agreement», ρ, will be used to predict v i and to exploit the relationship between y ij and x i more efficiently.
The usual hedonic price model is specified by equation [1] making y i equal to the market price and assuming that the variability of y i is mainly due to measurable attributes x i , and hence the variability, V(y i ) = σ 2 v , of y i around its expected value, x T i β, is assumed to be low.It is also assumed that there is no autocorrelation among perturbations, Cov(y i , y i' ) = 0 for all i ≠ i' and that the attributes are measured without error.

The empirical best linear unbiased predictor
It is assumed that associated with the i th property there is a set of n i indicators {y ij ; j = 1.2,...,n i } of y i as well as a vector x i of measurable attributes of the said property.The set {(y ij , x i ); j = 1,2,...,n i ; i = 1,2,...,m} can be observed for a sample of m properties.The consensus estimator proposed to estimate y i is the empirical best linear unbiased predictor (EBLUP), y ˆi, based on model [3]: where β ˆis the empirical generalized least squared estimator (EGLS) of β, v ˆi = g i (y ¯i -x T i β ˆ) is the predictor of the contextual factors effect, and Due to the fact that the two error components, (v i , e ij ), induce autocorrelation among the perturbations in [3], the ordinary least square (OLS) estimator of β is inefficient: the more efficient estimator is the generalized least square (GLS).However, the GLS estimator is not feasible because it depends on the unknown variance components, σ 2 v and σ 2 e .The EGLS used in [4] is the GLS once σ 2 v and σ 2 e have been replaced by consistent estimators (see Appendix for details).The usual hedonic price model assumes that there is no autocorrelation among perturbations and, under this assumption, OLS and GLS coincide.
The EBLUP has optimal statistical properties (Goldberger, 1962;Robinson, 1991).However, it is necessary to verify the assumptions of the model since the statistical properties of the EBLUP are optimal only if these assumptions are correct.A statistical test for this predictor is given in the Appendix.Equation [4] is defined as an estimator and not an indicator of value, given that it is possible to measure its standard variation.To this end the mean squared error of the empirical predictor, MS ˆE(y ˆi), proposed by Prasad and Rao (1990) (also see Ghosh and Rao, 1994) will be used, and the standard error of y ˆi will be estimated as the square root of MS ˆE(y ˆi).
The prediction of the effect due to unobserved contextual factors, v ˆi = g i (y ¯i -x T i β ˆ), is proportional to the residual value (y ¯i -x T i β ˆ), which is a result of the subtraction from the indicator's mean, y ¯i, of the component due to the measurable attributes x T i β ˆ.The coefficient of proportionality, g i , depends on both the variability due to contextual factors, σ 2 v , and the agreement among indicators,σ 2 e .When there is total agreement among the indicators, then σ 2 e = 0 and g i = 1 and the whole residual value is estimated to be due to contextual factors.When there is a certain disagreement among indicators then σ 2 e > 0 and only a proportion 0 <g i < 1 of the residual value is due to the contextual factors; the remainder is considered to be due to the disparity in the perception of β.When σ 2 v approaches zero, the effect of contextual factors tends to zero.

Relative efficiency of the empirical best linear unbiased predictor estimator
The estimator in [4] can be written as y ˆi = g i y ¯i + + (1 -g i )x T i β ˆ, that is a weighted mean of the extreme estimators y ˆi (0) = x T i β ˆand y ˆi (1) = y ¯i, corresponding to g i = 0 and to g i = 1, respectively, with the weights being g i and (1 -g i ).The value g i = 0 corresponds to the case where it is assumed that the variability of the property value due to unobserved contextual factors is null (σ 2 v = 0), and the whole variability is due to measurable attributes and the value g i = 1 corresponds to the case where it is assumed that the whole variability of the observed properties value is due to unobserved contextual factors.
The estimator y ˆ(0) is known as the synthetic regression estimator in statistics literature.When the hedonic assumption holds (σ 2 v → 0), then y ˆi (0) coincide with the usual hedonic estimator and this is why it is called hedonic regression estimator from now on.When the hedonic assumption holds, then the hedonic regression estimator approaches the EBLUP.However, when the hedonic assumption does not hold and the variability of the real properties value is mainly due to the contextual factors effect, then the EBLUP approaches the mean estimator, y ˆi(1) = y ¯i.

Results: A case study
A sample of m = 38 agricultural properties from Castilla y León (Spain) is considered, and a set of five indicators {y ij ; j = 1,2,3,4,5} of the value of each agricultural property is given.The land of the agricultural properties in the sample is currently being used for barley crops or vineyards, either using unirrigated or irrigated systems.The attributes of value measured in each agricultural property are the size of the property (x 1i ), the distance of the property both to the nearest road (x 2i ) and to the nearest town (x 3i ), and the net income per hectare (x 4i ).The choice of attributes is inspired by the economic rational of the three usual approaches to the valuation problem as well as in the land valuation literature (Platinga and Miller, 2001).The net income (gross income minus expenses, without taking opportunity costs into account) is calculated assuming that the technical and economic conditions are those corresponding to the average farm in the area.
One of the five value indicators (i) is the valuation by an expert appraiser operating in the studied area and the other four were calculated using common methods: (ii) the so-called rental capitalization approach, defined as the present discounted value of the rate of rental, (iii) the so-called income capitalization approach, defined as the present value of the net incomes, (iv) a valuation based on the ratio between the valuation of the expert and the crop net income, which consists in multiplying the observed ratio in the set of properties with a same crop and under the same system (unirrigated or irrigated) by the yield of the crop observed in the property, and, finally, (v) a variant of this last method, consisting in using the mean of individual ratios, instead of the ratio of the set.
Table 1 shows the attributes of the average agricultural property in the sample.Agriculture in Castilla y León is similar to that of many regions and it is expected that the attributes collected in Table 1 will also be similar and that the empirical results arising from this article will also be of use in regions which are similar to the Spanish region mentioned above.Furthermore, in order to broaden the usefulness of results, the procedure suggested has been applied to a wide range of data simulated on the basis of the model adjusted to observed data.
The F-Fisher & Snedecor test for the difference between means of indicators of value across the same crop type under a similar system (unirrigated or irrigated) does not reject the equality of means hypothesis.This result supports the assumption that the indicators are unbiased.
Land use is considered as explanatory to the value of a property, and a dummy variable is associated with each observed category of land use observed in the sample.The significance of the eight variables considered as explanatory to value (four dummy and four attributes) is tested using a Wald test and the estimates β ˆand Varβ ˆdef ined in the Appendix [A.1].The net income per hectare and the dummy variables associated with unirrigated barley, irrigated vineyards and unirrigated vineyards are statistically significant at the 1% level, while the remaining four variables considered are not significant at the 5% level, possibly due to the low variability among properties in the sample, as shown in Table 1.
Table 2 shows the estimates of β and (in brackets) the estimates of the standard error of the estimators, together with [in square brackets] the Wald statistics results, for the significant variables.
Table 3 shows the estimates of the variance components.
The estimation of variability due to the contextual factors' effect is σ ˆ2 v = 894,951.25 and the estimation of variability due to differences between indicators and difficulties involved in perceiving both β and v i , as well as the measurement errors in the attributes is σ ˆ2 e = 6,606,256.9.The «level of agreement» between indicators is estimated to be ρ ˆ= 1 -σ ˆ2 e /(σ ˆ2 v + σ ˆ2 e ) = 0.12, so that there is a notable disagreement, 1 -ρ ˆ= 0.88, among the value indicators used.
The value of λ lm observed when using [A.3] (see appendix) is 4.23 and since χ 2 0.95 (1) = 3.84, the null hypothesis, σ 2 v = 0, is rejected at a 5% signif icance level.Hence, it is admitted that g i differs from zero, so that the EBLUP differs from both the hedonic regression and the arithmetic mean of the estimators.

Relative efficiency between estimators
Table 4 shows the EBLUPs value per hectare of the sample properties, using expression [4], and the typical standard error of the EBLUP (square root of the estimated mean squared error of the estimator computed by using (A.4) (see appendix)], as well as the remaining estimators.The EBLUP is the most precise (the lowest standard error), while the worst estimator is the indicators' mean arithmetic.
The relative efficiency of the EBLUP with regard to the arithmetic mean ranges between 1.64 and 1.83 (e.g. the variance of the arithmetic mean is 64% to 83% higher than the mean square error of the EBLUP estimator), and with regard to the hedonic regression estimator, ranges between 1.17 and 1.47.

Parameterization of the «degree of agreement» and the contextual factors effect
The relative efficiency of the EBLUP with regard to the hedonic regression and arithmetic mean estimators, depend on the ratio between the variance components σ 2 v and σ 2 e .In order to study this dependency, both the «degree of agreement» among indicators, σ 2 e , and the variability due to contextual factors, σ 2 v , were parameterized, assigning ρ values between 0.1 and 0.9, with an increase of 0.1.The values of σ 2 v y σ 2 e , corresponding to each of ρ's values have been calculated, with the restriction that the total variation should be the same as that observed in the sample.
A set of values {y ij ; j = 1,2,3,4,5; i = 1,2,...,38} for each parameterized value of ρ were simulated using model [3] and the data relative to value attributes observed in the sample.These simulations have been calculated adding the value of v i , simulated from a normal variable of mean zero and variance σ 2 v and the values {e ij ; j = 1,2,3,4,5}, simulated from a normal variable of mean zero and variance σ 2 e to x T i β ˆ.
The relative efficiency of EBLUP with regard to both hedonic regression and arithmetic mean estimators calculated taking the simulated values is shown in Figure 1.In order to represent the graph, a box-andwhiskers graph has been associated with each of the values assigned to ρ.The box length is the inter-quar-  tiles range of relative efficiency, the point within the box marks the median and the «whiskers» mark the maximum and minimum relative efficiencies.As expected, EBLUP is the most efficient (its relative efficiency is higher than 1) for every ρ value.When ρ is low (whether the reason for this is that the level of disagreement among indicators is high with regard to the variability due to contextual factors or that the aforementioned variability is low with regard to the disagreement), then EBLUP is notably more efficient than the arithmetic mean and only slightly more efficient than the hedonic regression estimator.On the contrary, when ρ is high (whether the reason is because the degree of agreement among indicators is high with respect to variability due to contextual factors or because this variability is high with regard to the level of

Discussion
There is a disagreement in the literature over the best method to measure the value of real estates, and this has traditionally been a serious obstacle, in particular for the valuation of agricultural properties.The methods proposed to date provide indicators of a property value, but not the standard error of these indicators and -as a result-there is no way to know which method is the most precise.To estimate the value of a given real property, the IVSC suggests using three different approaches: the sales comparison approach, the income capitalization approach and the cost approach.The problem is that the appraiser has to make a final reconciliation step to reach a f inal estimate of the market value and there is no existing methodology on how to combine the different market values calculated by the three approaches into one final market value.
The statistical approach proposed in this paper provides the best linear combination of value indicators provided by the methods of valuation proposed in the literature, including the hedonic approach, and is, thus, the best compromise among these methods.The consensus estimator proposed uses all available information optimally, including the information which value indi-cators contain on the effect of contextual factors.This furnishes it with a greater relative efficiency with regard to alternative estimators, such as the arithmetic mean of the indicators or the hedonic regression estimator.While the former solely uses the indicators' data -ignoring data relative to attributes-the latter uses data relative to attributes but does not use data relative to indicators and assumes that the effect of contextual factors is negligible.
A MIMCM model is estimated, using data on a sample of agricultural properties in Castilla y León (Spain), and the relative efficiency of the consensus estimator was assessed using this model.The empirical results confirm that the proposed estimator is more efficient than the alternative linear estimators.The inefficiency of these alternative estimators depend on both, the level of disagreement between indicators and the contextual factors' effect, which has been evaluated using simulation.When the disagreement among indicators is high with respect to the contextual factors' effect, then the proposed estimator is notably more efficient than the arithmetic mean of indicators, although only slightly more efficient than the hedonic regression estimator.However, when a high level of agreement among indicators is achieved or the contextual factors' effect is high with respect to the level of agreement among indicators, then the consensus estimator proposed is only slightly more efficient than the arithmetic mean of indicators, but much more efficient than the hedonic regression estimator.

The best linear unbiased predictor of y i
The BLUP of y i based on model [3] is taken from Goldberger (1962) and Robinson (1991) and has many applications (Ambrosio and Iglesias, 2000): where 1 is a column vector (n × 1) of ones and x the matrix (n × p) of the attribute values, ) is a block diagonal matrix with , where I (n i ) is the identity matrix of order n i and 1 (n i ) is a column vector (n × 1) of ones.The covariance matrix of β ˆis Var β ˆ=

Testing model assumptions
In the specified model, the basic assumption is that y i is not uniquely determined by the measurable attributes, x i , but also by unobserved contextual factors, which makes y i vary around its expected value, x T i β, with variance σ 2 v > 0. The null hypothesis σ 2 v = 0

Relative efficiency (i) With regard to the arithmetic mean estimator
The mean estimator, , does not make use of attributes data, only of indicators y ij .The variance of this estimator is given by V [y ˆi (1)] = S 2 i /n i , where S 2 i is the variance between indicators of a same property.Assuming that S i 2 is the same in every property and equal to S w 2 , it can be estimated by where .

(ii) With regard to the hedonic regression estimator
The relative efficiency of the EBLUP with regard to the hedonic regression estimator would be estimated by M ˆSE[y ˆi (0)]/M ˆSE(y ˆi).If the hedonic assumption holds and the contextual effect is low in such a way that σ 2 v /σ 2 e is small, then g i tends to be low so that on the basis of [A.4] M ˆSE[y ˆi (0)]/M ˆSE(y ˆi) tend towards 1 and the hedonic regression estimator tends to be as eff icient as the EBLUP.When both the contextual effect and the agreement among indicators are high enough for σ 2 v /σ 2 e to be high, then g i is far from zero and the BLUP tends towards the estimator, y ˆi (1), so that V ˆ(y ˆi)/M ˆSE(y ˆi) tends towards 1 and the mean estimator tends to be as efficient as the EBLUP.

Estimation when indicators are not available
When the set of indicators is not available, then [4] is not feasible and it is proposed to use x T i β ˆas estimator of y i and to estimate its mean squared error using MS ˆE(y ˆi) = x i (X T V ˆ-1 X ) -1 x T i + σ ˆ2v .

Figure 1 .
Figure 1.The relative efficiency of EBLUP with regard to both hedonic regression and mean arithmetic estimators.

Table 1 .
Attributes of the average agricultural property

Table 2 .
Estimates coefficients ): standard error of the estimates coefficients.[]: Wald statistics of the estimates coefficients. (

Table 3 .
Estimates of the variance components

Table 4 .
Estimates based on EBLUP and relative efficiencies assuming that σ 2 v and σ 2 e are known), whereg i = σ 2 v /[σ 2 v + (σ 2 e /n i )] andY is the column vector (n × 1) of the y ij values in the sample.Substituting the BLUP in v ˆi in [A.1], y i 's BLUP becomes: