Estimation of Hedonic Price Functions via Additive ... - CiteSeerX

Estimation of Hedonic Price Functions via Additive Nonparametric Regression by Carlos Martins-Filho Department of Economics Oregon State University Ballard Hall 303 Corvallis, OR 97331-3612 USA email: [email protected] Voice: + 1 541 737 1476 Fax: + 1 541 737 5917

and

Okmyung Bin Department of Economics East Carolina University A-426 Brewster Greenville, NC 27858-4353 USA email: [email protected] Voice: + 1 252 328 2872 Fax: + 1 252 328 6743

November, 2001

abstract . We model a hedonic price function for housing as an additive nonparametric regression. Estimation is done via a back¯tting procedure in combination with a local polynomial estimator. It avoids the pitfalls of an unrestricted nonparametric estimator, such as slow convergence rates and the curse of dimensionality. Bandwidths are chosen via a novel plug in method that minimizes the asymptotic mean average squared error (AMASE) of the regression. We compare our results to an alternative parametric model and ¯nd evidence of the superiority of our nonparametric model. From an empirical perspective our study is interesting in that the e®ects on housing prices of a series of environmental characteristics are modeled in the regression. We ¯nd these characteristics to be important in the determination of housing prices. Keywords and Phrases: additive nonparametric regression; local polynomial estimation; hedonic price models; housing markets. JEL classi¯cations. C14, R21.

1

1

Introduction

Hedonic price models have been used extensively in applied economics since the seminal work by Rosen(1974).1 A frequent concern in this literature is the adequacy of commonly assumed parametric speci¯cations as hedonic price functions. This speci¯cation problem arises quite naturally from the inability of economic theory to provide guidance on how characteristics of similar products relate functionally to their market prices. Recognizing the potentially serious consequences of functional misspeci¯cation, most researchers have attempted to estimate hedonic price models by specifying more °exible regression models. Most of these attempts have concentrated on parametric speci¯cations ranging from simple data transformations, including the model introduced by Box and Cox(1964) and its variants, approximations based on second order Taylor-expansions to °exible non-linear models such as those introduced by Wooldridge(1992). A considerably smaller set of authors have proposed semi and fully (unrestricted) nonparametric speci¯cations for hedonic price functions. Nonparametric regression models are very °exible in that regressions are allowed to belong to a vastly broader class of functions than that in parametric models. However, their use in applied economics has not been as prevalent as one would expect, or comparable to their use in other disciplines, such as Biostatistics.2 In fact, nonparametric estimation of hedonic price functions has appeared in just a few articles and has concentrated almost exclusively in housing markets. Among these papers are the important contributions of Hartog and Bierens(1989), Stock(1991), Pace(1993,1995), Anglin and Gencay (1996), Gencay and Yang (1996) and Iwata, Murao and Wang(2000). Closer inspection of this literature, however, reveals the possibility of a number of methodological improvements that can add to both the ease of obtaining, and interpreting hedonic price nonparametric functional estimates. These improvements fall into three broad categories: (a) speci¯cation of the regression class; (b) choice of the smoother underlying the estimation; and (c) choice of the bandwidths. In this paper we address each of these categories. With the exception of Iwata, Murao and Wang(2000), all the articles listed above specify a regression class 1 Recent

examples include among many Dreyfus and Viscusi(1995), Walls (1996) and Hill, Knight and Sirmans (1997). provides a list of potential reasons for this relative scarcity of applied nonparametric economic modeling.

2 Yatchew(1998)

2

that requires the estimation of multivariate smoothers. There are a number of practical, as well as theoretical problems that emerge when estimating multivariate smoothers. First, there is the curse of dimensionality identi¯ed by Friedman and Stuetzel(1981). The problem is specially acute in data sets used by economists in hedonic price function estimation. These data sets are not in general very large but have at times a vast list of product attributes or characteristics which contribute to slow convergence rates of the estimators and diminished con¯dence on inference. Second, when de¯ning neighborhoods in two or more dimensions for local averaging, an universal characteristic of multivariate nonparametric estimation, there is the need to assume some type of metric that is hard to justify when the variables are measured in di®erent units or are highly correlated. Third, from a practical perspective, multivariate smoothers are extremely expensive to compute and even with the use of sophisticated graphical analysis four or higher dimensional smooths are virtually impossible to represent or interpret. Even in more parsimonious speci¯cations, such as that in Robinson(1988) and Speckman(1988), applied research inevitably encounters the uncertain prospect of ¯tting multivariate smoothers. Since one of the objectives of hedonic price modeling is to easily interpret and isolate the contributions of a given attribute to market price variability, holding all other product characteristics ¯xed, we ¯nd the use of a fully unrestricted nonparametric regression undesirable. We believe much better results in hedonic price modeling can be obtained by estimating an additive nonparametric regression model (ANRM), as in Hastie and Tibshirani(1986). Estimation of these models involve only univariate smoothing but allow for multiple regressors, and due to their additive nature lend themselves to easy interpretation and analysis. Regarding the choice of estimator, we depart from the standard literature in applied economics by using local polynomial estimators rather than the popular Nadaraya-Watson (NW) estimator. Recent work by Fan(1992), Fan, Gasser, Gijbels, Brockmann and Engel(1993) and Ruppert and Wand(1994) has shown that local polynomial estimators possesses a number of desirable theoretical and practical properties relative to other smoothing methods, including the NW estimator. One of the most serious pitfalls of the NW estimator

3

that we are able to avoid is its sensitivity to boundary data points. Lastly, we believe we make signi¯cant improvements over previous applied work by estimating bandwidths via a plug in method rather than the popular but much criticized cross validation.3 Although plug in methods are relatively new in the statistical literature they have consistently outperformed cross validation as a data driven bandwidth selection method. They converge at faster rates, are less expensive to compute and overcome the problem of undersmoothing that is characteristic of the cross validation method.4 In this paper we use data from the Portland-Oregon housing market to estimate a hedonic price function using all of the improvements described above. Since the estimation procedure is new and computationally intense, we produced an easy to use computer code that allows for fast implementation of the procedure, making it readily available for applied researchers. From an empirical perspective our model is of independent interest as a series of environmental and locational housing characteristics such as property elevation, distance to wetlands, parks and lakes are included in the hedonic price function. We also compare our nonparametric regression results to those that are obtained from a parametric speci¯cation based on a second order Taylor's approximation. The rest of this paper is organized as follows: in section 2 we give some generalities on the additive nonparametric model and describe the estimation strategy and its properties; section 3 describes and summarizes the data that we have used in this study; section 4 speci¯es the empirical model, gives details on the computational aspects of the estimation algorithm and introduces a viable parametric alternative; section 5 presents and analyzes the results and provides an out-of-sample forecast exercise. Section 6 is the conclusion.

2

Model Speci¯cation and Estimation

2.1

Generalities

We model hedonic price functions as a multivariate regression model given by, 3 See 4 See

Park and Marron(1990) and Simono®(1996). Sheather and Jones(1991) and Ruppert, Sheather and Wand(1995).

4

E(Y jX1 = x1 ; ¢ ¢ ¢ ; XD = xD ) = ® +

D X

md (xd ):

(1)

d=1

We assume that n independent observations f(yt ; xt1 ; :::; xtD )0 gnt=1 are taken on the random vector (Y; X1 ; ¢ ¢ ¢ ; XD ) and that V (Y jX1 = x1 ; ¢ ¢ ¢ ; XD = xD ) = ¾ 2 , an unknown parameter. The md (¢) are real valued measurable functions with E(md (¢)) = 0, E(m2d (¢)) < 1 belonging to Hd a Hilbert space with inner product given by E(md (Xd )m0d (Xd )) for each d.5 Under these assumptions E(Y ) = ® and the optimal predictor for Y given X1 ; ¢ ¢ ¢ ; XD can be characterized by 0

md (xd ) = E @Y ¡ ® ¡

D X

±=1;±6 =d

1

m± (¢)jXd = xd A

(2)

for d = 1; :::; D.6 The system of equations in (2) provides the motivation for the back¯tting estimator (B-estimator) for md (¢) proposed by Friedman and Stuetzle(1981). The idea is to simultaneously solve (2), estimating the conditional expectations via a suitably chosen linear smoother with regressand given by Y ¡®¡

PD

±=1;±6 =d

m± (¢) and regressor given by Xd . A precise de¯nition of the B-estimator requires some

additional notation. Let an ´ (a; ¢ ¢ ¢ ; a)0 be the n £ 1 vector with the constant a as its components, In

¡ ¢ be the identity matrix of size n, Dn ´ In ¡ n¡1 1n 10n , ~y ´ (y1 ; :::; yn )0 , ~xd ´ (x1d ; :::; xnd )0 , m ~ d (~xd ) ´

(md (x1d ); :::md (xnd ))0 , Sd be the n £ n smoother matrix associated with regressor d and Sd¤ ´ Dn Sd be a ~ d (~xd ) is the solution for the following system of centered smoother for d = 1; ¢ ¢ ¢ ; D.7 The B-estimator for m normal equations, 0

¢ ¢ ¢ S1¤ B ¢ ¢ ¢ S2¤ B B . .. @ . .. ¤ ¤ SD SD ¢ ¢ ¢ In ¡ b ¢0 which we denote by m ~ 1 (~x1 ); ¢ ¢ ¢ ; m ~ bD (~xD ) . In S2¤ .. .

S1¤ In .. .

5 m0 (¢) denotes an element of the H which d d 6 See Buja, Hastie and Tibshirani(1989). 7 The

10 CB CB CB A@

m ~ 1 (~x1 ) m ~ 2 (~x2 ) .. . m ~ D (~xD )

1

0

C B C B C=B A @

S1¤ S2¤ .. . ¤ SD

1

C C C ~y; A

is di®erent from md (¢).

speci¯c form for Sd depends on the univariate nonparametric estimator used in (2).

5

(3)

A convenient procedure to obtain a solution for (3) is to use the Gauss-Seidel algorithm. In practice, its implementation involves setting initial values ®0 ´ n¡1 10n ~y and m ~ bd (~xd )0 ´ 0n for all d. We then de¯ne the v th iteration (v = 1; 2; ¢ ¢ ¢) estimator m ~ bd (~xd )v as the smooth that results from a suitably chosen nonparametric univariate regression estimator, where the observed regressands are given by

y ¡ 1n ® 0 ¡ ~

d¡1 X ±=1

m ~ b± (~x± )v ¡ 0

D X

±=d+1

Iterations continue until jj~y ¡ 1n ® ¡

PD

m ~ b± (~x± )v¡1 and the regressors are ~xd , for d = 1; ¢ ¢ ¢ ; D.

±=1

m ~ b± (~x± )v jj22 ¡ jj~y ¡ 1n ®0 ¡

than a pre speci¯ed level of tolerance, where if µ 2 0 for all x 2 (ad ; bd ); A3. As n ! 1, hdn ! 0 and nhdn ! 1 for all d; A4. The (p + 1)th derivatives of md (¢), d = 1; :::; D exist and are continuous. One of the practical conveniences of the local polynomial estimators is that provided that p is large enough (q)

and that md (¢) is su±ciently smooth, the q th derivative of md (¢), denoted by md (x) can be estimated by,

b(q) md (x)

=

0

0 0 ¡1 q!ep+1 Rd (x)0 Wd (x) @y q+1 (Rd (x) Wd (x)Rd (x))

b(q)

and therefore we can de¯ne, m ~d

2.2

b(q)

¡ 1n ®0 ¡

D X

±=1;±6 =d

1

m ~ b (~x± )A

(5)

b(q)

(~xd ) = (md (x1d ); ¢ ¢ ¢ ; md (xnd ))0 .

Data Driven Selection of hdn

One of the most important steps in estimating any nonparametric regression model is the choice of hdn . In essence, after a nonparametric estimation procedure is chosen the selection of the bandwidths is tantamount to the selection of an estimated regression. It is therefore desirable to use a data driven method for their selection. The most commonly used procedure, especially in applied econometrics, involves choosing h1n ; h2n ; ¢ ¢ ¢ ; hDn that minimize a jackknifed sum of squares or cross validation function. Unfortunately, besides being extremely computationally intense, cross validation has a tendency to undersmooth producing estimated regressions that have rather high variance. Furthermore, the di®erence between the cross validation estimator of hdn for a local polynomial estimator and the hdn that minimizes the estimator's mean 8 See

Opsomer(1995) and Martins-Filho(2001).

7

average squared error (MASE) converges to zero at a very slow rate. These undesirable characteristics, have prompted the use of plug in methods for bandwidth selection. The basic principle behind plug in methods is the direct estimation of functionals that appear on the expressions describing the bandwidths that minimizes the function used to guide the choice of the estimated regression. Speci¯cally, we choose ~hn ´ (h1n ; ¢ ¢ ¢ ; hDn )0 2 0

(8)

K(u) with Np a (p + 1) £ (p + 1) matrix having (i; j)

ui+j¡2 K(u)du, Mr;p is the same as Np except that the r + 1 column is substituted by

(1; u; u2 ; :::; up )0 , and det(A) is the determinant of the squared matrix A. The other component of (7) that needs to be estimated is ¾ 2 . We de¯ne, ¾ ^ 2 (~·n ) = n¡1 jj~ y ¡ 1n ®0 ¡

PD

d=1

m ~ bd (~xd )jj22 as a generic estimator for ¾ 2 based on ~·n 2 0

(9)

The expressions for the optimal bandwith vectors in (7), (8) and (9) depend on unknown functionals, µdd (2; 4), µdd (2; 2) and ¾ 2 . Our estimation strategy follows Opsomer and Ruppert(1998) and is summarized as follows. We obtain pilot estimates for ¾ 2 and µdd (2; 4) for d = 1; :::; D based on a suitably de¯ned parametric model. These initial estimates are used to compute g^dn according to equation (8). We then use g^dn as bandwiths to ¯t an additive model using a polynomial smoother of degreee p = 3. We then use the ¯tted additive model to estimate µ^dd (2; 2). This estimate together with the initial estimate of ¾ 2 are then ^ dn are then used to ¯t an additive model using a local polynomial of used to obtain · ^dn according to (9). · order p = 1 and to compute an updated estimate of ¾ 2 . Finally, the updated estimate of ¾ 2 together with 9

^ dn according to (7). The h ^ dn are then used to obtain the estimate of µ^dd (2; 2) are used to obtain estimates h a ¯nal ¯t for the ANRM and ¾ 2 . Before we proceed to describe and analyze the data, a few comments are in order regarding our chosen estimation procedure. First, Linton and HÄ ardle(1996) have proposed an alternative estimator for the ANRM called the marginal integration (MI) estimator. We chose not to use the MI estimator for two reasons: a) the algorithm required to produce the estimator calls for the estimation of a multivariate fully nonparametric regression model of order D. With the relatively large number of regressors in hedonic price models, the curse of dimensionality becomes a major concern; b) Martins-Filho(2001) provides experimental evidence that the ¯nite sample performance of the back¯tting estimator is superior to that of the MI estimator. Second, the asymptotic and ¯nite sample distibutions of the B and MI estimators are currently unknown when ~hn is chosen by any data driven method.10 However, following Hastie and Tibshirani(1990) we are able to construct approximate con¯dence intervals for the B-estimators.

3

The Data

The housing market data that we use comes from a portion of Multnomah County, Oregon-USA that lies within Portland's urban growth boundary. The regressand in our empirical model is the actual recorded sales price of a dwelling. We use 1000 recorded sales that occured between June of 1992 and May of 1994 in the study area. All sales prices were adjusted to May 1994 levels using a Multnomah County residential housing market price index. Figure 1 provides a histogram of housing prices with a bandwith chosen by the method proposed by Sheather and Jones(1991). Direct inspection of the histogram reveals that the vast majority of the observations falls within the (50000,250000) interval. There is however a great dispersion of sales prices and some observations are quite distanced within this interval. Another clear conclusion from the histogram is the leptokurdic nature of the distribution. Table 1 provides a list of all variables used in our study as well as some sample statistics. The regressors can be grouped into two main categories. The 10 This

is also true for the MI estimator.

10

¯rst contains a series of dwelling attributes commonly used in hedonic price studies, such as dwelling area, land area, the total number of bedrooms and bathrooms, as well as the dwelling age in 1994. The second contains a series of locational characteristics of the dwelling. They include distances to the central business district, the nearest lake, improved park, industrial sector, commercial district and wetland. Also included is a measure dwelling elevation. All linear measurements are in kilometers, and all area measurements are in squared meters.11 The data come from two major sources; MetroScan, which compiles real estate data from assessor's records for numerous U.S. cities provided most of the house's structural attributes data; Metro Regional Services, a regional government agency provided the locational characteristics. All distance calculations were made using a raster system where all data are arranged in grid cells. Each cell is a 15.6-meter square with distances measured using the Euclidean norm from the center of the dwelling land parcel to the nearest edge of the feature. Wetland locations are based on the U.S. Fish and Wildlife Service's National Wetlands Inventory in Oregon. Wetlands vary from primarily open water to forest and grassland that is wet only part of the year. Although the area of urban wetlands has been declining in the U.S. the section of Multnomah county included in our study has more than 4500 wetlands and deep water habitats, varying in size from 1 to 358 acres.12 . We have some a priori beliefs regarding the e®ects of these various characteristics on housing prices. Speci¯cally, we expect a positive association between sales prices and the dwelling's area, number of bathrooms, land area, the dwelling's elevation and the distance to the nearest industrial zone. Conversely, we expect a negative association between sales prices and distances to lakes, parks and the central business district as well as the dwellings age. We have no a priori expectations regarding the e®ects of wetland distance on sales prices. Although proximity to wetlands may be perceived as a desirable location characteristic due 11 Squared

meters can be converted to squared feet dividing by 0.093 and kilometers can be converted to miles dividing by

1.6. 12 See

Mahan, Polasky and Adams(1998).

11

to enhanced view quality or increased pollution protection, it can also be undesirable due to increased risk of °ooding or wildlife annoyances. We are also unsure about the e®ects of proximity to commercial districts on house prices. Although close proximity may be undesirable due to increased tra±c and noise, large distances maybe undesirable due to increased transportation costs. It is important to point out that our a priori expectations are from a descriptive point of view rather crude. One of the advantages of departing from a linear parametric model involves the possibility of unveiling much richer patterns of association between regressand and regressors. Put di®erently, the standard practice in applied economic work of revealing expected parameter signs is in itself evidence of how restricted linear regression parametric modeling can be.

4

Empirical Modeling and Computations

Since two of the variables used as regressors are discrete (number of bathrooms and bedrooms), we estimate the following semiparametric version of (1),

E(Y jX1 = xt1 ; ¢ ¢ ¢ ; X12 = xtD ) = ® +

2 X d=1

¯d xtd +

12 X

md (xtd )

(10)

d=3

for t = 1; :::; n and Y and X1 ; ¢ ¢ ¢ ; X12 as indicated in Table 1.13 The estimation procedure requires initial estimates for ¾ 2 and µdd (2; 4). The latter requires estimates (4)

(2)

for md (¢) and md (¢). To obtain these initial values we ¯rst ¯t a linear parametric regression model that results from a Taylor's expansion of order 5 that explores the additivity of the conditional expectation of Y including all regressors that appear in (10). This estimated regression provides initial estimates for the second and fourth derivatives of md , which are then used to construct the ¯rst estimates of µdd (2; 4). These, together with an estimate for the variance ¾ 2 are used in equation (8) to obtain ~g^, which is given in Table 2. ~g^ is then used to ¯t (10) using the B-estimator and a local polynomial estimator of order p = 3. From this 13 The data, as well as the computer code for the implementation of the estimation procedure, which was written in the GAUSS v.3.2.14 (1996) programming environment, are available from the ¯rst author.

12

¯t of the additive model we obtain a new estimate for the second derivatives of md using (5). These new estimates of the second derivatives are used to obtain an updated estimate for µdd (2; 2) which together with ^ n reported in Table 2 is then used to the variance estimate are used to obtain · ^ dn according to (9). The ~· ¯t (10) once again, so that a new estimate for ¾ 2 is obtained. This new variance estimate is used with the ^ ^ updated estimate of µdd (2; 2) to obtain ~hn according to (7) and reported in Table 2. Finally, ~hn is used to obtain a ¯nal ¯t of (10), which is then used to obtain a ¯nal estimate for ¾ 2 . We denote these ¯nal estimates ^2. by ®b , ¯1b , ¯2b , mbd (~xd ) for d = 3; ¢ ¢ ¢ ; 12, and ¾ We have a series of observations to make regarding the estimation procedure. First, the fact that the estimated bandwiths depends on ~y produces estimators that are not projectors and are nonlinear. However, as desired, the cycles of the back¯tting algorithm produced a decreasing sequence of jj~y ¡ 1n ®0 ¡

PD

±=1

m ~ b± (~x± )v jj22 .

Convergence was obtained for the ¯rst ¯tting of the ANRM after 13 cycles, for the second ¯tting after 11 cycles, and for the ¯nal ¯tting convergence was attained after 9 cycles, all with a level of tolerance of 0.001. With n = 1000 the computation time for the estimates was approximately 4 hours and 33 minutes on a 866 Mhz Pentium III PC. The estimated regressions are the solid lines that appear on Figure 2. In the next section we provide a detailed discussion of the results. Second, one major di±culty in ¯tting an ANRM via back¯tting (or marginal integration) with data driven estimated bandwiths, is that we are unable to construct asymptotically valid con¯dence intervals for the estimated bandwiths. It is possible however to obtain an estimated covariance matrix for each mbd (~xd ) for d = 1; ¢ ¢ ¢ ; 12. This results from the fact that at convergence, mbd (~xd ) can be written as Rd ~y for ^ 2 Rd R0d . We obtain Rd by noting that at some n £ n matrix Rd . Hence, V (mbd (~xd )) can be estimated by ¾ convergence it can be calculated by the following numerical procedure. We de¯ne the identity matrix of size ^ n = 1000 to be In and denote its ith column by I:i . Using ~hn we ¯t the ANRM in (10) with regressand given ~ bd (~xd ) that correspond to the ith column of Rd , for d = 1; ¢ ¢ ¢ ; n. by I:i . This produces n-dimensional vectors m Hence, we run 1000 new ANRM to obtain Rd . The dashed lines that appear in Figure 2 are lower and upper

13

bounds on the estimated regressions constructed by multiplying by 2 the square root of the diagonals of ¾ ^ 2 Rd Rd0 . In all of these 1000 auxiliary models, convergence was attained in less than 6 cycles. Although these con¯dence bands are not exact, they provide what we believe are very suitable approximations.14 Third, our ¯nal variance estimator is obtained according to the following expression ¾ ^2 = ¯1b ~x1 ¡ ¯2b ~x2 ¡

P12

d=3

mbd (~xd )jj22 , where R ´

P12

d=1

1 y¡ n¡tr(2R¡RR0 ) jj~

Rd and tr(¢) is the trace operator. The purpose is to ac-

count for the degrees of freedom (df ) inherent in our estimation procedure. For our data we obtained df = 828. Although this is smaller degrees than the degrees of freedom of a quadratic parametric approximation (df = 909), it is much larger than what would result if an approximation of order 3 were used. Put simply, our nonparametric estimator does not seem to require an unreasonable amount of degrees of freedom. The ¯nal variance estimate for the ANRM is ¾ ^ 2 = 9:39 £ 108 . It is instructive to compare our results to a linear parametric speci¯cation for this hedonic price function. More speci¯cally, we would like to provide some comparison between what researchers would normally propose as a parametric model of regression for a hedonic price model and our speci¯cation. We guided our choice for the parametric model by the following constraints: a) we consider only models for E(Y jX1 ; ¢ ¢ ¢ ; XD ), ruling out speci¯cations such as E(T (Y )jX1 ; ¢ ¢ ¢ ; XD ), where T (¢) is some transformation of Y . This rules out models that are well known to be ill-speci¯ed, such as the models proposed by Box and Cox (1964), as well as popular semi-log (T (Y ) = log(Y )) models. We do so because we concur with Wooldridge(1992) in that our interest is on the conditional expectation of Y not a transformation of Y . Such transformations produce ambiguities on the analysis and interpretations of the regression results; b) we assume that the researcher has no a priori knowledge about the speci¯c nature of E(Y jX1 ; ¢ ¢ ¢ ; XD ) as to permit the estimation of a well de¯ned nonlinear regression model. Within these constraints, we speci¯ed and estimated the parametric regression model given by (11), 14 See Hastie and Tibshirani(1990). We remind the reader that even if the asymptoic distribution of the B-estimator were available, the con¯dence bands would still be approximations.

14

E(Y jX1 = xt1 ; ¢ ¢ ¢ ; XD = xtD ) = ® +

D X

D

µd xtd +

d=1

D

1 XX µd± xtd xt± where µd± = µ±d : 2

(11)

d=1 ±=1

Similar parametric second order approximations have been extensively used in applied econometrics. Table 3 provides ordinary least squares estimates for the parameters in (11), as well as some other commonly reported regression statistics. The parametric model provides a very reasonable ¯t for the data. We observe that model (11) does not have the additivity property of the ANRM. If one is interested on the relationship between Y and Xd , ceteris paribus, it is necessary to arbitrarily choose values for all X± where ± 6 = d to characterize such relationship. Choosing average sample values for other regressors is common practice in the empirical literature. We observe, however, that this procedure generates simply one of (uncountably) many potential estimated regressions. The same can be said about ¯rst derivatives and elasticities that derive from these models. We view this as a major inconvenience of this type of parametric approximation. For comparison with the ANRM we have estimated each regression direction based on the parametric model with all other regressors evaluated at their averages. Their estimated price e®ects are the dash-dotted lines that appear on Figure 2. Additivity is an important assumption of the ANRM, hence we performed a test for additivity proposed by Hastie and Tibshirani(1990). The test involves estimating the following arti¯cial regression,

rt =

D X X

Ãdj mbd (xtd )mbj (xtj );

(12)

d=1 j>d

where rt = yt ¡ ®b ¡

P2

d=1

xtd ¯db ¡

P12

d=3

mbd (xtd ) are the residuals from the estimated additive model. A

conventional Student's-t statistic for H0 : Ãdj = 0 against HA : Ãdj 6 = 0 is calculated. We interpret rejection of H0 as evidence that the interaction of regressors is strong enough to reject additivity. Conversely, failing to reject H0 lends support to the additivity assumption. Almost all t-values lead to non-rejection of H0 at 1 percent, therefore supporting the additivity assumption. 15 Results

15

are available upon request. Interaction terms can easily be handled in an additive non parametric framework

15

5

Estimation Results and Analysis

5.1

Parametric Estimates

Our parametric model is simple and for the casual observer may seem rather inappropriate, especially given the availability of much richer parametric speci¯cations.16 However, after informal experimentation with several alternative parametric speci¯cations that do not expand the set of regressors, we were surprised to learn that the best overall ¯t resulted from our model. Our informal search is by no means evidence that one cannot ¯t a better parametric model, it is simply indicative that (11) does reasonably well against some obvious parametric alternatives. The results in Table 3 suggest that the most important attributes in determining sales prices are the dwelling area, the lot square footage and dwelling age. Among the locational attributes the most signi¯cant determinants of sales prices are the dwelling's elevation, its distance to the central business district, nearest wetland and nearest commercial district. The results are reasonable and generally in line with comparable previous studies, e.g., Iwata et al.(2000). We will make more speci¯c comments regarding the estimated parametric model as we contrast its results with that of the ANRM. All of the variables used in the estimation of (11) are in deviation form. Hence, the parametric model can be interpreted as a second order Taylor's approximation around the sample average of the regressors.

5.2

Nonparametric Estimates

We start our analysis by observing that the estimated regressions have very resonable shapes. No clear instance of over or under smoothing is apparent and the fact that a common bandwith was used in each regression direction did not create problems for most of the regressors' data range. There seems to be some similarities between the regression estimates produced by the parametric model and the ANRM for some PD PK by specifying, for example, E(Y jX1 = xt1 ; ¢ ¢ ¢ ; XD = xtD ) = ® + md (xtd ) + mk (ztk ), where ztk ´ xtd xt± for d=1 k=1 d6 = ±; and d; ± = 1; ¢ ¢ ¢ ; D. 16 One could try to improve the parametric ¯t by expanding the number of regressors. An almost surely successful strategy would be to add sets of dummy variables for the regressors involving distances.

16

regressors, e.g., dwelling area and land area, however for most regressors the two models reveal signi¯cantly di®erent impact on sales price. These are more pronounced for the impact of distance to the nearest lake, industrial zone, central business district and dwelling elevation. We now turn to a more speci¯c analysis of each estimated regression. In various occasions we will refer to data ranges where most of the observations fall. These can be promptly seen in Figure 3, where the estimated nonparmetric regressions are graphed together with the n = 1000 additive partial residuals. Dwelling and land area: The estimated regressions for the in°uence of a dwelling's area and land area on price have very reasonable shape, with the exception of the more variable behavior of mb3 for X3 > 234.17 We attribute this variability to undersmoothing in this data range and place little credence on the sharp increase on the estimated regression in that data range. In the range (60; 234), where most of the data is concentrated, mb3 is increasing and nearly linear, with a ¯rst derivative that oscillates between 300 and 1,000 dollars. The estimated regression is slightly convex for dwellings with area larger than 180, indicating a larger impact of size on prices for larger dwellings. For a dwelling that is +3:5 standard deviations from average the impact of additional area is about 3; 000 dollars or more. It is not surprising, given the near linearity of mb3 , that the parametric model produces results that are very similar to those suggested by mb3 . As we expected mb4 suggests, in general, a positive association between sales price and land area. We place little weight on the declining portion of the estimated regression for lots whose area is below 227. It corresponds to a data range where there are only 9 observations. The impact on sales prices of land area is much less pronounced than that of dwelling. Within §1 standard deviation (134; 1; 320) from the average land size (727) sales prices vary by less than 60 dollars for a marginal increase in land area. The results from the parametric model are similar in that the estimated regression has positive slope, but the impact of land area on sales prices is even smaller than that suggested by the nonparametric model. We believe that in this case, even though m4 is nearly linear, the parametric model underestimates the impact of land area on 17 Since all regressors are centered, this corresponds to the centered X being greater than 100. All of our analysis of the 3 estimated results will be made using noncentered values for the regressors.

17

prices. This results from the fact that the OLS estimator of the parametric model is highly in°uenced by the observations in the (827; 1; 000) range, which have a much smaller impact on the nonparametric estimator of m4 where most of the data lies. Dwellinig age: The impact of dwelling age on sales prices is di±cult to intepret. The data seems to indicate that housing developments has occured in fairly well de¯ned phases. We were able to identify four data clusters corresponding to the periods 1920 ¡ 1930, 1945 ¡ 1956, 1970 ¡ 1981, and 1991 ¡ 1994. The fact that the data is clustered has produced a bandwith that seems to undersmooth mb5 , hence the wiggly appearence of the estimated regression. We are able, however to discern some patterns. Dwelling age impacts prices negatively in the (1975; 1990), i.e., for house with age between 4 and 18 years. Ceteris paribus a house built in 1990 will cost about 20; 000 dollars more than one built in 1975. However, for dwellings that are between 20 and 80 years old the impact of age on sales prices seems to be much less pronounced. In fact, not counting the 1935-1945 period, for dwellings between 20 and 80 years old, the impact of age on sales prices is smaller than §10; 000 dollars. The sharp increase in mb5 from 1935 ¡ 1945 seems to be caused by a combination of sample variability and undersmoothing. The results that derive from the parametric model are quite di®erent. Age according to (11) has little impact on sales prices for dwellings up to approximately 35 years old. After that the negative impact of age intensi¯es continuously. An 80 year old dwelling is expected, ceteris paribus to sell for about 25; 000 dollars less than a 50 year old. Distance to the nearest lake: Close proximity to a lake has a substantial positive impact on sales prices. However, this impact is restricted to dwellings that are less than 2 kilometers away from a lake. Here, the impact on prices from moving away from a lake is substantial. Ceteris paribus, moving a house from 2 to 4 kilometers away from a lake produces a drop in sales price of approximately 30; 000 dollars. The in°uence of proximity to a lake on sales price falls dramatically after a dwelling is more than 3 kilometers away. In fact, m5 (¢) ´ 0 falls within the con¯dence band around mb5 almost everywhere. These results are very intutitive. After a certain distance use of the lake for recreational purposes is limited and even prohibited by

18

some neighborhood associations (in the case of private lakes) and transportation costs start to outweight the bene¯ts of lake use. The parametric model produces an altogether di®erent and unappealing interpretation of the data. First, the drop in sales prices as one moves away from a lake is very gradual and decreasing. The sharp drop in prices captured by mb5 at short distances from a lake is not revealed by the parametric model. Second, and more disturbing, the parametric model predicts an increase in sales prices for dwellings that are more than 6 kilometers away from a lake. Distance to the nearest wetland: There is a negative relationship between sales prices and distance to the nearest wetland for most of the data range. This negative impact is more pronounced within §1 standard deviation of the average distance (1.09 kilometers). Moving a dwelling adjacent to a wetland 2 kilometers away produces a decrease in price of about 20; 000 dollars. Althought this e®ect is smaller than the lake e®ect, it is signi¯cant and suggests a market that incorporates quite well the value of environmental amenities. As in the case with distance to a lake, dwellings that are farther away from a wetland (in this case more than 2 kilometers) have prices that are impacted little by wetland distance. Note that the impact of wetland distance on sales prices dies out more rapidly than the impact o f distance to a lake. The parametric model was able to predict well the impact of wetland distance in sales prices. The estimated parametric relationship between these two variables falls within the con¯dence bands we have constructed for mb7 for almost all the data range. Distance to the nearest improved park: mb8 reveals that in general proximity to an improved park has little impact on sales prices. In fact, m8 (¢) ´ 0 falls within the con¯dence band we constructed for virtually all data range. Price variations where most of the data lies (:1; :6) kilometers is all within §3000 dollars. The parametric model produces similar results. Dwelling elevation: mb9 predicts a positive impact of dwelling elevation on sales prices for the interval (20; 100) meters. The gain in price for this data range is 40; 000 dollars. After 100 meters mb9 levels out with a derivative that is very close to zero. For elevations above 200 meters the impact on sales prices

19

increases sharply, however as in previous cases there are just a few observations on this data range. It seems reasonable to expect a positive association between elevation and prices, due to the possibility of views and quieter streets. However, higher elevations may also be associated with higher transportation costs. Hence, the fact that mb9 levels o® is expected and intuitively appealing. The parametric model produces quite di®erent predictions. Dwelling prices ¯rst drop (until about 60 meters) with elevation then rise continuously at an increasing rate. Clearly, there is no cost associated with elevation gain. According to the parametric model, a house at average elevation (81 meters) has a price increase of almost 80; 000 dollars when it gains 100 meters in elevation. Distance to the nearest industrial, commercial zones and the central business district: mb10 clearly captures the negative impact on sales prices of dwellings that are close to industrial zones. Furthermore, it indicates that this negative e®ect is specially intense very close to the industrial zone. For example, moving a dwelling that is adjacent to an industrial zone 400 meters away, increases its value ceteris paribus by approximately 20; 000 dollars. The bene¯ts from moving away from industrial zones diminish at an increasing rate. Interestingly, there seems to be a negative impact on sales prices after about 2 kilometers. It is likely that this results from increased transportation costs to the work place. Once again the predictions of the parametric model are quite di®erent. Prices are predicted to fall as we move away from industrial zones (until about .7 kilometers). For distances above .7 kilometers, prices increase continuously at an increasing rate. The fall in prices due to increased transportation costs captured by the nonparametric model is not revealed by the parametric model. mb11 suggets a positive relationship between sales prices and distances to the nearest commercial zone. The impact however is much smaller than that of X10 . That is, the problems of congestion, pollution, etc. associated with proximity to a commercial zone do not seem as intense as those related to proximity to industrial zones. As in the case of mb10 we observe that after a certain distance, there is a reversal of the ¯rst derivative sign, and distance begins to have a negative impact on sales prices. Once again, we atribute this

20

e®ect to a predominance of transportation costs over congestion costs. Here, the parametric model predicts better than in the case of X10 , however the negative impact of distance in prices at the upper range of the data is not captured. Proximity to the central business district (CBD) has a very strong positive impact on sales prices for the ¯rst 3 kilometers. Ceteris paribus moving a dwelling that is adjacent to the CBD to a location 3 kilometers away may reduce its price by as much as 170; 000 dollars. However, while still negative this e®ect is reduced dramatically after 3 kilometers. For example, moving from 3 kilometers to 9 kilometers away from the CBD impacts price by only 20; 000. This is in line with our general expectation. The e®ect of distance to the CBD for the parametric and nonparametric models are quite di®erent. Once again, the very sharp initial decline on prices as a typical dwelling is moved away from the CBD is largely unaccounted by the parametric model. Overall, we believe that the nonparametric results are more appealing than those suggested by the parametric model. It comes as no surprise that whenever the estimated regressions are close to linear the parametric model provides very adequate responses. However, even in this case estimated slopes can be over or under estimated depending on the pattern of dispersion of the data, as in the case of land area.

5.3

Out-of-Sample Forecast Exercise

One of the characteristics of hedonic price models is that they can be used to forecast prices, given a set of product characteristics. We were able to perform a simple out-of-sample forecast evaluation of both the paprametric and nonparametric models based on a new sample of 1000 observations, which we denote by N n 18 f(ytN ; xN The new sample comes from the same housing market and corresponds to the t1 ; ¢ ¢ ¢ ; xt12 )gt=1 .

same regressors and regressand. Using the new data, we obtained forecasted sales prices for the parametric model, 18 We have re-estimated the ANRM based on n = 2000 and the results are virtually identical to those reported here for n = 1000. Results are available from the ¯rst author upon request.

21

y^tp = ® ^+

D X d=1

D

D

1 XX ^ N N µ^d xN µd± xtd xt± ; where µ^d± and µ^d are the least squares estimators. td + 2 d=1 ±=1

and for the ANRM,

y^tb

b

=® +

2 X

¯db xN td

d=1

+

12 X

mbd (xN td ):

d=3

^ We used ~hn from Table 2 as the bandwith used in the estimation of the ANRM and we did not update ®b . Using fytN gnt=1 we calculate the square root of the average squared forecast error for the parametric (F E p ) and ANRM (F E b ) estimators to be F E p = 31251:19 and F E b = 27897:84. Hence, the forecast error of the ANRM is about .89 of that corresponding to the parametric model. This di®erence can translate into substantial di®erences. For example, assuming a property tax rate of 0:014 of dwelling market value, this represents an average 47 dollars tax adjustment per dwelling. Assuming 200; 000 tax units, this represents about 9:4 million dollars in property tax adjustments.

6

Conclusions

In this paper we have argued that the functional form speci¯cation problem common in hedonic price models can be conveniently addressed by modeling the conditional mean of prices as an additive nonparametric regression model. The approach is in our view vastly superior to a fully nonparametric design for several reasons. First, from a practical perspective, the smooths are very easy to interpret and visualize permitting therefore an analysis of the contribution of characteristics and attributes to prices in much the same way that is obtained in classical separable parametric models. Second, from a statistical perspective, a series of well known di±culties of an unrestricted fully nonparametric design are avoided. Our bandwith selection method is novel, entirely data driven, and has performed well in pactice vis a vis the popular cross validation selection method. To facilitate the implementation of the method among applied economists we have written a GAUSS program that permits relatively fast implementation of the procedure. 22

Most importantly, after implementing the procedure using data for Multnomah County, Oregon, we verify that the nonparametric additive model is able to identify associations and patterns of dependencies among the data that a reasonably adequate parametric model fails to unveil. Besides, an out-of-sample evaluation of the forecast errors of both the parametric and nonparametric models reveals a superior performance of the ANPRM. We are optimistic that the econometric modeling strategy used in this study can be successfully used in a variety of settings. Despite our optimism, we ¯nd that the estimation procedure used in this paper needs to be understood better. We are speci¯cally concerned with our current inability to construct more precise con¯dence intervals and perform some rather standard tests of hypotheses.

7

References 1. Anglin, P. and R. Gencay (1996) Semiparametric Estimation of Hedonic Price Functions, Journal of Applied Econometrics, 11, 633-648. 2. Box, G.E.P and D.R. Cox (1964) An Analysis of Transformations, Journal of the Royal Statistical Society B,2 26, 211-252. 3. Buja, A., Hastie, T.J. and R.J. Tibshirani (1989) Linear Smoothers and Additive Models, Annals of Statistics, 17, 453-555. 4. Dreyfus, M.K. and W. Kip Viscusi (1995) Rates of Time Preference and Consumer Valuations of Automobile Safety and Fuel E±ciencyJournal of Law and Economics, 38, 79-105. 5. Fan, J. (1992) Design Adaptive Nonparametric Regression, Journal of the American Statistical Association,8 87, 998-1004. 6. Fan,J., T. Gasser, I. Gijbels, M. Brockmann, J. Engel(1993) Local Polynomial Fitting: a Standard for Nonparametric Regression. Department of Statistics, UNC. 7. Friedman, J.H. and W. Stuetzel (1981) Projection Pursuit Regression, Journal of the American Statistical Association,7 76, 817-823. 8. Gencay, R. and X. Yang (1996) A Forecast Comparison of Residential Housing Prices by Parametric and Semiparametric Conditional Mean Estimators, Economic Letters, 52, 129-135. 9. Hartog, J. and H. Bierens(1991) Estimating a Hedonic Earnings Function with a Nonparametric Method, in A. Ullah ed.Semiparametric and Nonparametric Econometrics: Studies in Empirical Economics,Springer Verlag-Heidelberg.

10. Hastie, T.J. and R.J. Tibshirani (1986) Generalized Additive Models, Statistical Science, 1, 297-318. 11. Hastie, T.J. and R.J. Tibshirani (1990) Generalized Additive Models, Chapman and Hall-New York. 23

12. Hill, R.C., J.R. Knight and C.F. Sirmans (1997) Estimating Capital Asset Pricing Indexes Review of Economics and Statistics, 79, 226-233. 13. Iwata, S., Murao, H. and Q. Wang (2000) Nonparametric Assessment of the E®ects of Neighborhood Land Uses on the Residential House Values, in T. Fomby and R. Carter Hill Eds. Advances in Econometrics: Applying Kernel and Nonparametric Estimation to Economic Topics, v.14, JAI Press. 14. Linton, O. and W. HÄ ardle (1996) Estimation of Additive Regression Models with Known Links, Biometrika, 82, 93-100. 15. Martins-Filho, C. (2001) Additive Nonparametric Regression Estimation via Back¯tting and Marginal Integration: Experimental Finite Sample Performance. Working Paper Department of Economics, Oregon State University. 16. Mahan, B., S. Polasky and R. Adams (2001) Valuing Urban Wetlands: A Property Price Approach Land Economics,7 76 , 100-113. 17. Opsomer, J. (1995) Optimal Bandwith Selection for Fitting an Additive Model by Local Polynomial Regression. Ph.D. Thesis Cornell University. 18. Opsomer, J. (1999) Asymptotic Properties of Back¯tting Estimators. Working Paper Department of Statistics, Iowa State University. 19. Opsomer, J. and D. Ruppert(1997) Fitting a Bivariate Additive Model by Local Polynomial Regression, Annals of Statistics,2 25, 186-211. 20. Opsomer, J. and D. Ruppert(1998) A Fully Automated Bandwidth Selection Method for Fitting Additive Models, Journal of the American Statistical Association,9 93, 605-619. 21. Pace, R. Kelley (1993) Nonparametric Methods with Applications to Hedonic Models, Journal of Real Estate Finance and Economics,7 7, 185-204. 22. Pace, R. Kelley (1995) Parametric, Semiparametric, and Nonparametric Estimation of Characteristics Values within Mass Assessment and Hedonic Pricing Models,Journal of Real Estate Finance and Economics,1 11, 195-217. 23. Park, B. and J.S. Marron (1990) Comparison of Data-Driven Bandwidth Selectors, Journal of the American Statistical Association,8 85, 66-72. 24. Robinson, P. (1988) Root-N Consistent Semiparametric Regression, Econometrica, 56, 931-954. 25. Rosen, S.(1974) Hedonic Prices and Implicit Markets: Product Di®erentiation in Pure Competition, Journal of Political Economy, 82, 34-55. 26. Ruppert, D., and M.P. Wand (1994) Multivariate Locally Weighted Least Squares Regression, The Annals of Statistics,2 22, 1346-1370. 27. Ruppert, D., S.J. Sheather, and M.P. Wand (1995) An E®ective Bandwidth Selector for Least Squares Regression, Journal of the American Statistical Association,9 90, 1257-1270. 28. Sheather, S.J. and M.C. Jones (1991) A Reliable Data-based Bandwidth Selection Method for Kernel Density Estimation, Journal of the Royal Statistical Society B,5 53, 683-690. 29. Simono®, J.S. (1996) Smoothing Methods in Statistics, Springer-New York. 24

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25

Table 1 - Sample Statistics Average Standard Deviation Y Sale Price 126783.55 83676.478 X1 Number of bathrooms 1.432 0.632 X2 Number of bedrooms 2.863 0.938 X3 Dwelling area 134.021 56.864 X4 Land area 727.526 593.712 X5 Dwelling age in 1994 44.299 27.299 X6 Distance to nearest Lake 5.389 2.090 X7 Distance to the nearest wetland 1.092 0.755 X8 Distance to nearest improved park 0.397 0.269 X9 Dwelling elevation 0.081 0.043 X10 Distance to the nearest industrial zone 1.141 0.930 X11 Distance to the nearest commercial zone 0.382 0.320 X12 Distance to the central business district 9.541 5.340 Variable

Table 2 - Estimated Bandwiths ^ ^n ~h ~g^ ~· Variable n n

X3 X4 X5 X6 X7 X8 X9 X10 X11 X12

50.757 587.303 9.537 0.907 0.484 0.153 0.028 0.682 0.177 1.646

26

23.802 369.669 6.769 0.572 0.434 0.173 0.02 0.477 0.172 0.978

14.819 230.147 4.214 0.357 0.270 0.108 0.012 0.297 0.107 0.609

Maximum 857053.90 6 14 584.546 10117.141 106 10.437 3.466 1.688 0.269 4.648 2.327 23.711

Minimum 30308.630 1 0 40.134 100.335 0 0 0 0.016 0.003 0 0 1.287

Parameter Intercept µ1 µ2 µ3 µ4 µ5 µ6 µ7 µ8 µ9 µ10 µ11 µ12 µ1;1 µ2;2 µ3;3 µ4;4 µ5;5 µ6;6 µ7;7 µ8;8 µ9;9 µ10;10 µ11;11 µ12;12 µ1;2 µ1;3 µ1;4 µ1;5 µ1;6 µ1;7 µ1;8 µ1;9 µ1;10 µ1;11 µ1;12 µ2;3 µ2;4 µ2;5 µ2;6 µ2;7 µ2;8 µ2;9 µ2;10 µ2;11 µ2;12

Table 3 - Parametric Model Estimates Estimate t-Statistic Parameter Estimate -16110.30 -4.99 µ3;4 0.18 -2204.54 -0.67 µ3;5 -7.83 -2428.11 -1.34 µ3;6 -69.79 751.14 20.17 µ3;7 24.87 18.43 3.77 µ3;8 -21.56 -400.75 -5.54 µ3;9 278.65 -1493.64 -1.96 µ3;10 -143.31 -10384.36 -4.16 µ3;11 397.00 920.78 0.17 µ3;12 -60.73 271890.14 4.18 µ4;5 0.51 3601.94 1.43 µ4;6 0.99 17728.83 3 µ4;7 -0.58 -5610.88 -10.07 µ4;8 -32.86 20934.33 3.17 µ4;9 -140.48 330.88 0.35 µ4;10 2.12 1.08 1.37 µ4;11 -2.50 0 -0.06 µ4;12 1.01 -10.44 -2.14 µ5;6 -52.76 1638.07 2.77 µ5;7 -12.67 5323.58 1.12 µ5;8 130.77 32805.12 1.51 µ5;9 -462.50 14135246 8.50 µ5;10 -31.87 10765.51 2.50 µ5;11 179.90 55163.05 2.65 µ5;12 -59.53 350.06 2.70 µ6;7 -389.99 11418.46 3.52 µ6;8 5094.98 -241.22 -4.34 µ6;9 -16491.79 -0.1 -0.02 µ6;10 4354.75 22.30 0.15 µ6;11 -10418.81 3131.92 2.20 µ6;12 98.61 -2542.19 -0.69 µ7;8 3391.70 7787.23 0.81 µ7;9 71561.56 -40265.17 -0.57 µ7;10 -687.72 5049.34 1.47 µ7;11 -7024.37 -20385.22 -2.31 µ7;12 -796.44 93.12 0.15 µ8;9 123569.33 -21.45 -0.67 µ8;10 5964.38 6.00 1.55 µ8;11 -8433.39 145.89 1.63 µ8;12 1026.29 -267.74 -0.30 µ9;10 -655593.01 154.44 0.06 µ9;11 -417165.62 8961.74 1.35 µ9;12 -52892.09 17932.93 0.35 µ10;11 -7381.84 -3544.23 -1.44 µ10;12 -21.13 2425.61 0.40 µ11;12 873.31 571.27 1.15 ¾ ^ 2 = 1:07 £ 109 df = 91:00 27

t-Statistic 2.47 -4.59 -4.05 0.55 -0.17 0.33 -3.27 4.05 -7.27 2.68 0.55 -0.09 -2.78 -1.60 0.43 -0.23 1.37 -1.64 -0.13 0.54 -0.22 -0.35 0.64 -2.99 -0.29 2.0 -0.63 3.13 -4.14 0.50 0.46 0.72 -0.21 -0.80 -1.25 0.81 0.76 -0.50 0.84 -8.52 -3.07 -4.90 -1.01 -0.04 0.69 R2 = 0:86

Figure 1

28

Figure 2

29

Figure 2

30

Figure 2

31

Figure 2

32

Figure 2

33

Figure 3

34

Figure 3

35

Figure 3

36

Figure 3

37

Figure 3

38