# Integrating the equation-based and adjustment-based approaches to transactions evidence timber appraisal.

U.S. Department of Agriculture (USDA) Forest Service timber appraisers find themselves in a dilemma. Recent federal legislation and agency directives generally required that field units adopt transactions evidence appraisal (TEA) by October 1992.(1) Unfortunately, however, some types of TEA are not accurate, while others are difficult to understand.

In TEA, timber value is estimated directly from market information derived from past timber sales. Two variations predominate: the adjustment-based approach and the equation-based approach. While the adjustment-based approach is intuitively meaningful, it can be quite inaccurate. Conversely, the equation-based approach is comparatively accurate, but many find its mathematical rigor incomprehensible.

In the adjustment-based approach, the value of an "average" timber sale is calculated (usually as the mean of recent sales) along with the mean levels of several sale features, such as the volume harvested and the final product value. The new timber sale is initially appraised at the overall mean of recent sales, then adjusted to reflect how that sale's features differ from the average sale. This procedure is easily understood by timber appraisers. Unfortunately, when the authors compared the accuracy of six timber appraisal methods in predicting timber value, the adjustment-based approach came in dead last.(2)

The equation-based approach uses a multiple linear regression model to estimate the value of a timber sale based on information from sale characteristics.(3) When comparing the six appraisal methods, the authors found this approach highly accurate.(4) Unfortunately, appraisers who lack extensive training in multiple regression analysis find timber appraisal equations mystifying.

In this article, we show how to integrate the two approaches to TEA, capitalizing on both the inherent logic of the adjustment-based approach and the inherent accuracy of the equation-based approach.

THE PROBLEM

The appeal of the adjustment-based approach lies in the simplicity of using an average or mean value (i.e., the value of the average sale) and then adjusting that average, based on the unique features of the sale being appraised. The average value is a concrete number to which appraisers can relate. The inaccuracy of this approach, however, results from the quality of adjustments. Adjustments are typically highly subjective and are rarely based on rigorous, quantitative analysis. In fact, the authors found that use of the adjustment-based approach was frequently less accurate than simply appraising a timber sale on the basis of the average value.(5) That is, timber appraisals were worse after adjustments were made.

The appeal of the equation-based approach lies in the rigorous manner in which the coefficients (i.e., the counterparts to adjustments) are estimated. The ordinary least squares (OLS) estimation technique commonly used in regression analysis finds the set of coefficients that minimize errors in estimation. The incomprehensibility of the equation-based approach results largely from attempts to interpret the coefficients, especially the intercept. In statistical terms, the intercept is the value of the response variable when predictor variables are set at zero. Some appraisers find it difficult to comprehend the meaning of a negative intercept (meaning a negative timber value) in the context of a timber sale with no volume and no final product value, because such conditions cannot exist in reality.

THE INTEGRATION

The equation-based approach to TEA is a multiple linear regression model defined in matrix notation as follows:(6)

Y = X|beta~ + |epsilon~

where the response variable (Y) is an n x 1 vector of observations (timber values), X is an n x p matrix of fixed, known values to be used as predictor variables (e.g., volume harvested, final product value, percent defect), |beta~ is a p x 1 vector of unknown parameters (i.e., coefficients) to be estimated, and |epsilon~ is an n x 1 vector of normally distributed random errors with a mean of zero and constant variance.

In the case of one predictor variable, the OLS technique simultaneously finds the intercept (a) and the slope coefficient (b) of the line that minimizes the sum of the squared differences between the line and the datapoints. It should be noted that the regression line passes through the point defined by the mean of Y (or |Mathematical Expression Omitted~) and the mean of X (or |Mathematical Expression Omitted~). In effect the regression line pivots at about that point. This principle extends to multiple variables; that is, the regression line, plane, or hyperplane identified by OLS will always pass through the intersection defined by |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, and so on.

Converting a standard regression model to reflect the adjustment-based approach simply requires that the Y-axis be shifted along the X-axis to |Mathematical Expression Omitted~. Figure 2a displays the same regression relationship as Figure 1, but with the Y-axis shifted. The new intercept is |Mathematical Expression Omitted~ (i.e., the value of Y when X is set to |Mathematical Expression Omitted~), but the regression coefficient remains the same. Conversion is completed by subtracting |Mathematical Expression Omitted~ from all points on the X-axis. The conversion recalibrates the X-axis, setting |Mathematical Expression Omitted~, shown in Figure 2b. The X-axis is now expressed in deviations from the average value of X. This modification to the standard regression model can be termed "centering the data";(7) we call it a "deviations-from-the-mean" form of regression.

The relationship displayed in Figure 2b can be derived in two ways. The first way uses the standard X-matrix, solves for the regression coefficients, and then shifts the intercept to |Mathematical Expression Omitted~ and recalibrates the X-axis:

given that

Y = a + bX

and

|Mathematical Expression Omitted~

substituting, we obtain

|Mathematical Expression Omitted~ |Mathematical Expression Omitted~

The second way modifies the standard X-matrix by subtracting |Mathematical Expression Omitted~ from all X values and solves for the regression equation:

then

|Mathematical Expression Omitted~

and because

|Mathematical Expression Omitted~

substituting, we obtain

|Mathematical Expression Omitted~

The procedures for converting a simple linear regression model to an adjustment-based approach extend directly to multiple linear regression. A regression problem can either be solved with original X data and converted later, or the X-matrix can be modified to solve for a deviations-based regression problem. Regardless, all standard statistical tests and problems still apply (e.g., tests of significance, collinearity).

AN EXAMPLE

To illustrate this suggestion, these two procedures are used to convert a three-variable, timber appraisal equation applicable to the western appraisal zone of the northern region of the USDA Forest Service. That zone generally consists of national forests in western Montana as well as in the northern half of Idaho. The data consist of information on 706 timber sales occurring from April 1988 to March 1991.

Procedure 1

This procedure begins with the standard regression model, enters all X variables at their means to estimate the intercept, and recalibrates the X-axis. The initial regression equation estimated is as follows:

Y = 29.47 + 0.83|X.sub.1~ + 0.36|X.sub.2~ - 1.15|X.sub.3~

where

Y = Timber value (dollars per 1,000 board feet |MBF~)

|X.sub.1~ = Volume harvested (million BF)

|X.sub.2~ = Final product value (dollars per MBF)

|X.sub.3~ = Percent log defect (percentage, in whole numbers)

During in. this time, the average sale value (|Mathematical Expression Omitted~) was \$119.20 per MBF and the mean values for X variables were:

|Mathematical Expression Omitted~ |Mathematical Expression Omitted~ |Mathematical Expression Omitted~

Accordingly:

|Mathematical Expression Omitted~

Procedure 2

In this procedure, the average values for |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, and |Mathematical Expression Omitted~ are subtracted from their respective observed values in the data set before analysis begins. Then a conventional, multiple linear regression analysis is performed. The resulting equation is:

Y = 119.20 + 0.83(|X.sub.1~ - 2.66) + 0.36(|X.sub.2~ - 285.32) - 1.15(|X.sub.3~ - 14.0)

DISCUSSION

This article demonstrates how to integrate the inherent logic of the adjustment-based approach with the inherent accuracy of the equation-based approach. This involves modification of the standard regression model into a deviations-from-the-mean form of regression. The modified appraisal equation can be used directly in an equation-based approach. Alternatively, the equation can be packaged as a set of narrative instructions associated with the more traditional application of the adjustment-based approach. The intercept is used as the average value and the coefficients are used as adjustments to reflect how an individual sale differs from the average sale.

Regardless, both approaches start from |Mathematical Expression Omitted~, the average timber value of past timber sales. To timber appraisers in the field as well as outside observers of the timber appraisal process, this feature grounds the procedure in reality. First, a known average timber value should be adjusted based on how the sale at hand differs from the average sale. If the sale does not differ, it will be appraised as the intercept (|Mathematical Expression Omitted~), the average value for past sales. Further, because the equation's intercept will be intuitively meaningful, the unending debates regarding the correct meaning of the traditional intercept term will be precluded. With a meaningful intercept, there also will be far less tendency to microanalyze changes in coefficients (or adjustments) over time.

The northern region of the Forest Service has used an unmodified, equation-based approach to timber appraisal for the past five years. During this time, a steering committee consisting of agency officials and industry timber purchasers has struggled with the problem of comprehending traditional equations. When this integrated method was recently presented to them, eyes opened and smiles appeared. Their response was to ask, "Why did it take you so long to figure this out?"

Ervin G. Schuster, PhD, is a research forester and project leader of the economics research project, U.S. Department of Agriculture (USDA) Forest Service, Intermountain Research Station, Missoula, Montana. He received academic training in economics and forestry at the University of Minnesota and Iowa State University, where he earned a PhD. Mr. Schuster specializes in the economics of multiple-use management.

Michael J. Niccolucci is an economist with the economics research project at the USDA Forest Service in Missoula, Montana. He received both a BA and an MA from the University of Montana. Mr. Niccolucci specializes in econometrics, economics analysis of timber sales, and timber appraisal.

1. 5 C.F.R. Part 1320 (January 1989): 129-145; and David L. Hessel, "Implementing Transaction Evidence Appraisal," unpublished directive, U.S. Department of Agriculture Forest Service, Washington Office, December 1991.

2. Ervin G. Schuster and Michael J. Niccolucci, "Comparative Accuracy of Six Timber Appraisal Methods," The Appraisal Journal (January 1990): 96-108.

3. John A. Combes, Michael J. Niccolucci, and Ervin G. Schuster, "Stumpage Appraisal: TEA and the Northern Region," Forest Industries (October 1989): 18-20.

4. Schuster and Niccolucci, 96-108.

5. Ibid.

6. See G. S. Madella, Econometrics (New York: McGraw-Hill, 1977), 448-452.

7. See Norman Draper and Harry Smith, Applied Regression Analysis, 2d ed. (New York: John Wiley & Sons, 1981), 257-266.